Strong Evaporation-Condensation In Gas-Dust Mixture
A. P. Kryukov, V. Yu. Levashov, I. N. Shishkova
Moscow Power Engineering Institute, Moscow, Russia
Abstract. Evaporation-condensation problem concerning the strong-nonequilibrum gas flow through the system of
randomly distributed immovable dust particles is investigated. In present work the rise of dust particles size due to vapor
condensation on these particles surfaces and reduction of this size due to evaporation are taking into account. The
velocity distribution function of gas molecules is found from the direct numerical solution of the Boltzmann kinetic
equation. The approach for calculation of the distribution function transformation as a result of gas molecules and dust
particles interactions is used. Gas numerical density, temperature profiles and variation of dust particles sizes for the
different time moments are presented.
1. INTRODUCTION
In the present work we study strong-nonequilibrum of gas-dust mixtures flow by means of molecular-kinetic
theory methods. The substantial peculiarity of this problem among analogous molecular-kinetic problems is the
presence of condensation nucleuses in vapor or vapor-gas mixture bulk. The sizes and masses of these nucleuses
(microscopic droplets or dust) can be very larger than the molecule size and mass. In addition, the variation of
nucleuses masses and sizes is possible. Thus these values depend on the evaporation-condensation intensity. In [1-2]
method for calculation of the velocity distribution function transformation as a result of gas molecules and
immovable dust particles interactions was described. In paper [3] the approach [1-2] was used for condensation of
gas molecules on the dust particles surfaces research. In [4] influence of dust particles motion on parameters of
investigated processes have been studied. It was noted in [4] that the description of flows of gas-dust mixtures
should be made on the base of the Boltzmann kinetic equations system. The Boltzmann kinetic equations (BKE) in
one-dimensional non-steady statement for gas-dust mixture can be presented as the following system:
∂f g
∂f
+ ξ x g = J gg + J gp
∂x
∂t
,
∂f p
∂f p
= J pp + J pg
+ ξx
∂t
∂x
(1)
where Jgg - collision integral describing interactions between molecules of gas; Jgp - collision integral describing
interactions between molecules of gas and dust particles; Jpp – collision integral describing interactions between dust
particles; Jpg – collision integral describing interactions between dust particles and molecules of gas. As is well
known the difficulties of system (1) solution increase at the rise of the ratio of masses of dust particle and gas
molecule. Therefore, in [4] the algorithm of system (1) simplification has been discussed. To realize this way from
system (1) the second equation (equation for dust particle) is excluded and collision integral Jgp is replaced by
certain numerical procedure describing the reflection of gas molecules from solid particles. In contrast to [3] the rise
of dust particles size due to vapor condensation on these particles surfaces and reduction of this size due to
evaporation is taken into account in this paper.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
2. PROBLEM AND SOLUTION METHOD
Evaporation-condensation problem in volume filled by immovable dust particles is considered. The statement of this
problem is shown in Fig. 1. The evaporation of gas nitrogen takes place at surface x = 0. The surface x = 5 is
absolutely cryogenic: the gas is absorbed completely. In the initial moment time (t = 0) domain x = 0 – 5 between
two surfaces is occupied by immovable spherical dust particles. Calculations are made for dust numerical density
Np=1.22*1021 m-3 and diameter Dp=93.8*10-10m. It is assumed that gas temperature at the entry in flow region is
T0=300K and pressure is P0 = 20kPa. Further all densities and temperatures are given in relation to basis parameter T0, P0. The co-ordinate x is given in the mean free paths of vapour molecules at these parameters (l) and time t is
given in relation to l
RT0 , where R – gas (nitrogen) constant. The condensation of gas molecules on the surfaces
of dust particles takes place and as a result diameter of dust particles is increased. The coefficient of condensation
(β) is accepted 0, 0.1, 0.5, 1.
Gas and dust
evaporation
5
0
x
condensation
FIGURE 1. Schematic illustration of the problem.
At the second stage of the evaporation–condensation at the presence of dust particles the following problem has
been solved. It is assumed that evaporation process is realized on the dust particle surfaces in accordance with
diffuse scheme. Thus it is considered that vapor molecules emitted from the dust surface have the Maxwell
distribution function with temperature of this surface, known saturation density corresponding to this temperature
and zero flow velocity. Dust temperature is suggested to be equal to base (initial) temperature T0. No molecules are
emitted from boundary surfaces x = 0 and x = 5 (see Fig.1) and all molecules arriving at these surfaces are reflected
in accordance with diffuse scheme also. As initial condition the non-dimensional vapor density in whole domain is
equal to unity at t = 0 (nt=0 = 1).
As it was noted above the solution of problem is made on the base of the first equation (equation for gas) of
system (1) where collision integral Jgp is replaced by numerical procedure describing modification of velocity
distribution function of gas molecules as a result of gas-dust interactions. The algorithm of modification velocity
distribution function of gas molecules is presented below.
It is assumed inside this discrete model that gas molecules accept only those values of the velocities ξ k , which
n r of the all molecules n has been reflected by solid
n
particles. In such a manner velocity distribution function f k is composed of two parts: the invariable portion f k
are determined by velocity grid. During time-step ∆t only part
corresponding to gas molecules which has not been interacted with dust particles; the second part
f kr of the
distribution function which has been transformed as a result of the reflection. Index k refers to k-point in velocity
space
f k = f kn + f kr .
The quantity of gas molecules in volume until interacting with dust particles during time ∆t is given by the
formula
nkr =
1
N p nkπD*2 g∆t
4
(2)
where D* = D p + d , Dp – diameter of the dust particle, d – diameter of the gas molecule, g – absolute value of
relative velocity of movement for dust particle and gas molecule, Np – numerical density of the dust particles,
nk = f k ∆ξ 3 – numerical density of gas molecules having velocity ξ k . The function f kr can be determined on the
base of expression (2).
Reflection of the gas molecules from dust particles is described by diffuse scheme (an example). The molecules
which had identical velocity ξ k before collision will have different velocities after it. Fig. 2(a) illustrates the process
k
of the transformation of distribution function. The values of gas numerical densities for reflected gas molecules nref
are determined by non-penetration condition. In case when the coefficient of condensation β is equal zero (β=0) this
condition is given by the formula:
r
nrefk = nrk
ξk
RTp
2π
where Tp - temperature of dust particles. Correspondingly for case β≠0 (the condensation of gas molecules on the
dust particles takes place) the following expression is valid:
nrefk = (1 − β )nrk
r
ξk
RTp
2π
An analogous procedure should be done for each point of velocity grid.
The velocity distribution function for gas molecules after collisions with dust particles
r*
summing all reflected distribution function f n (ξ n ) obtained for each grid velocity point
f kr* is found as a result of
ξn :
f kr* = ∑ f nr* (ξ k )
M
(3)
n =1
where M - the total quantity of velocity grid points. Expression (3) shows that each group of molecules having
before collision the velocity
ξn
makes its own contribution to the determination of distribution function
f kr* after
collision (Fig 2(b)).
Calculation
f kr*
in accordance with the above-described algorithm gives the possibility to determine velocity
distribution function of gas molecules after interaction with dust particles:
f k* = f kn + f kr* .
To obtain the distribution function in a whole velocity space it is necessary to repeat this procedure for all
velocity grid points.
As was noted above the rise of dust particles sizes owing to gas molecules condensation takes place for case
when β≠0. Correspondingly in this case the diameter of dust particle depends on time and can be determined by the
following way:
1
3
3
D p (t ) = 2 (V + χVm )
 4π

where Vm – volume of the all gas molecules which have been condensed on the dust particles, V – volume of the
dust particles on the previous time step, χ – the value characterized friable nature of condensate.
The velocity distribution function for each flow point is found as a result of solving the first equation from
system (1) using the direct numerical solving method [5]. Macroparameters (density and temperature) are moments
of the distribution function and are determined by the integration over a three-dimensional space of molecules
velocities (see for example [6]).
Before interactions
n kr
After interactions
fr
fr*
f kr
n kr *
0
ξ
ξk
ξ
(a)
M
f
r
r
f *
f
ξ
ξnk
r
n
f kr * = ∑ f nr * ( ξ k )
n =1
f nr * (ξ k )
ξ
ξk
ξ
(b)
FIGURE 2. The velocity distribution function transformation at interaction of gas molecules with dust particles.
3. RESULTS AND DISCUSSION
Solution results of the problem of gas flow in the system of randomly distributed immovable spherical particles
are presented in Fig. 3-5. It was noted above the sizes of dust particles increase during gas condensation on these
particles. Condensate is formed as friable deposit. For example, all results are given for coefficient χ=9.5. Figure 3.
presents gas numerical density and temperature profiles for different time moments and β=0.5. The variation of dust
particles sizes for different times are presented in Figure 3. by dotted line. These results show that the sizes of dust
particles increase during gas condensation. For example, for time moment t=0 the diameter of all dust particles are
equal to 93.8*10-10m but for time moment t=37.05 the diameters of dust particles are modified. As it is obvious from
Fig.3 the diameters of dust particles are different for each point of investigation domain. For example, for time
t=37.05 the dust particles have sizes between 1.1*10-8m and 2.4*10-8 m and for time t=101.01 the domain is
occupied by dust particles with sizes between 1.25*10-8m and 4.0*10-8m.
The gas parameters distribution along OX is presented in Fig 3-4. The analysis of the results of these figures
gives the conclusion about influence of dust particles rise on the gas parameters. For example, for time t=37.05 (Fig.
3a) the values of gas numerical density in region x=0-1 are more than the same values for time t=100.01 (Fig. 3b)
approximately two times. Such behavior of density is determined by the several reasons.
One of them is the following. The gas molecules collide not only with gas molecules but with dust particles also.
It is necessary to note that the quantity of these collisions increases when the sizes of dust particles rise. As a result
of such collisions the part of gas molecules reflected toward a surface х=0 increases and the quantity of molecules
"reached" surface x=5 decreases.
t=100.01
t=37.05
2,6x10
-8
0,9
0,8
2,4x10
-8
0,8
0,7
2,2x10
-8
0,7
2,0x10
-8
0,6
1,8x10
-8
1,6x10
-8
1,4x10
-8
1,2x10
-8
0,1
1,0x10
-8
0,0
n, Tg
0,6
n
Tg
Dp
0,5
0,4
0,3
0,2
0,1
0,0
0
1
2
3
4
5
-8
4,5x10
Dp, m
-8
4,0x10
-8
n, Tg
Dp, m
0,9
3,5x10
0,5
0,4
n
Tg
3,0x10
Dp
2,5x10
-8
-8
0,3
-8
2,0x10
0,2
-8
1,5x10
-8
0
1
2
3
X
4
1,0x10
5
X
(a)
(b)
FIGURE 3. Dependence of gas density n, gas temperature Tg and diameter of dust particles Dp on co-ordinate for β=0.5.
t=100.01
-8
4,5x10
-8
4,0x10
-8
3,5x10
-8
3,0x10
-8
2,5x10
-8
0,3
2,0x10
-8
0,2
1,5x10
-8
0,1
1,0x10
-8
β
β
β
β
=
=
=
=
0
0.1
0.5
1
β=0
β = 0.1
β = 0.5
β=1
0,7
0,6
0,5
0,4
n
Dp,m
0,8
5,0x10
0,0
0
1
2
3
X
(a)
4
5
0
1
2
3
4
5
X
(b)
FIGURE 4. Dependence of dust particles diameter – Dp and gas density – n on coordinate for moment time t=100.01 and
different condensation coefficients.
The second reason is the following. The dust particles can be considered as a condensation centers for gas
molecules. The quantity of a gas decreases during condensation because gas transfers into solid deposit and as a
result of this process diameters of dust particles rise. The values of dust diameters are greater near the surface х=0
than beyond. Therefore as the value of collisions a gas - dust increases as the total dust surface condensation
increases, and the density of gas decreases.
The dependencies of gas densities and diameters of dust particles on coordinates are presented in the Fig 4a, 4b
for different condensation coefficients. These results show (see Fig 4b) that the densities of gas decrease for a case
when the value of condensation coefficient increases from 0 to 1. Such a kind of behavior can be explained by
different quantity of gas molecules that have been reflected after collisions with dust particles. For example, for a
case β→1 all gas molecules which collide with dust particles "adhere to" surfaces and quantity of reflected gas
molecules is approached to 0. Therefore the density of gas decreases. It should be noted that the quantity of
condensed gas molecules on the dust particles are determined by the sizes of these particles.
t=100.01
0 ,7
D p = c o n s t, β = 0 .5
D p ris e s , β = 0 .5
0 ,6
0 ,5
n
0 ,4
0 ,3
0 ,2
0 ,1
0 ,0
0
1
2
3
4
5
X
FIGURE 5. The influence of dust particle sizes rise on value of gas densities.
The influence of dust particle sizes rise on value of gas densities is seen from the results presented in Fig.5. Here
the results obtained without taking into account the rise of dust particles sizes is presented by dotted line. In this
Figure the results with taking into account the rise of dust particles sizes are shown by solid line. One can see from
this Figure that the results of calculation for different cases are differed in three times. Analysis of these results
shows that at a correct description the rise of dust particles sizes owing to vapor condensation on these particles
surfaces should be taken into account.
It is necessary to notice that the considered problem is non-stationary since the sizes of dust particles are changed
and distribution function of gas molecules are changed also.
12
ns= const =1.0
10
n
8
β=0.1
6
4
β=0.3
β=0.5
2
β=1.0
0
0,0
-4
2,0x10
-4
-4
4,0x10
6,0x10
-4
8,0x10
-3
1,0x10
np/n
FIGURE 6. Dependencies of gas density on quantity of dust particles for different condensation coefficients.
Solution results of the problem when the evaporation-condensation processes take place at surfaces of dust
particles are presented in Fig 6-9. It should be noted that in this problem no molecules are emitted from boundary
surfaces x = 0 and x = 5 (see Fig.1) and all molecules arriving at these surfaces are reflected in accordance with
diffuse scheme. In this statement at a time evolution of two opposite directed processes take a place. First of them is
vapor condensation on the dust particles with a prescribed value of condensation coefficient β. Second process is
evaporation from dust surface with a prescribed value of saturation density ns. Due to these processes for each set of
β and ns definite values of microscopic and macroscopic characteristics are established in steady stage. Among
different macroscopic characteristics vapor density (n ) and dust particle diameter (Dp)can be interested for different
purposes.
β = const = 1.0
1,1
1,0
ns=1.0
0,9
0,8
n
0,7
0,6
0,5
ns=0.5
0,4
0,3
ns=0.3
0,2
-4
-4
-4
-4
-4
-4
-4
-4
-4
-3
-3
1,0x10 2,0x10 3,0x10 4,0x10 5,0x10 6,0x10 7,0x10 8,0x10 9,0x10 1,0x10 1,1x10
np/n
FIGURE 7. Dependencies of gas density on quantity of dust particles for different ns
It is seen from Fig.6 that at β = 1 vapor density is equal to unity for all rations np/n. Such behavior is
understandable because in this case mass flux density at condensation on the dust surface is exactly equal to mass
flux density at evaporation from this interphase and therefore n = 1. At smaller values β evaporation process prevails
over condensation therefore vapor density n rises. For example at β = 0.3 and more enough np/n vapor density is
bigger about three times than n for complete condensation. Waiting and obvious information is given in Fig.7: vapor
densities differ in two times when corresponding values ns differ in two times also.
ns= const =1.0
100
95
90
85
10
Dp*10 , m
80
75
β =1.0
β =0.5
β =0.3
70
65
60
55
β =0.1
50
45
-4
-4
-4
-4
-4
-4
-4
-4
-4
-3
1,0x10 2,0x10 3,0x10 4,0x10 5,0x10 6,0x10 7,0x10 8,0x10 9,0x10 1,0x10
np/n
FIGURE 8. The influence of evaporation on results of dust particle diameters calculations for different condensation coefficients.
Figures 8 and 9 illustrate the influence of evaporation on the results of dust particle diameters calculations. And
in these calculations initial particle diameter is 93.8Å. One can see in Fig.8 that this value does not vary with np/n at
β = 1 due to balance of evaporation and condensation. At smaller β particle diameter decreases because evaporation
becomes stronger than condensation. Such reduction is largest at minimal np/n. At the np/n rise this decreasing is
reduced. Possible explanation of this tendency can be connected with mutual influence of molecular fluxes from
different particles at large np/n. It should be noted that in the above discussed calculations minimal diameter of
particle is prescribed as Dp = 50 Å. If such type diameter prescription is not made than at small β (weak
condensation) particle can disappear completely. The dependency Dp on np/n for β = 0.1 confirms this conclusion.
Not evident influence variation of saturation vapor density on Dp versus np/n dependency is shown in Fig. 9. One can
see a strong influence of evaporation reduction at small np/n presented by curve for ns = 0.5 and ns = 0.3.
Nevertheless at large np/n difference between curve for ns = 0.3, ns = 0.5 and ns = 1.0 is diminished very strongly.
We believe that possible reason for this trend can be found in the analysis of molecular fluxes form different
particles and corresponding interactions of these fluxes.
β = const = 1.0
106
104
100
10
Dp*10 , m
102
ns=0.3
98
ns=0.5
96
ns=1.0
94
-4
-4
-4
-4
-4
-4
-4
-4
-4
-3
1,0x10 2,0x10 3,0x10 4,0x10 5,0x10 6,0x10 7,0x10 8,0x10 9,0x10 1,0x10
np/n
FIGURE 9. The influence of evaporation on results of dust particle diameters calculations for different ns.
ACKNOWLEDGMENTS
This work is supported by the Russian Foundation for Basic Research, grant 02-02-81015 and in part grant 0015-96543.
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