Evaporation And Condensation On A Plane Condensed
Phase In The Basis Of Discrete Kinetic Theory
Ioana Nicodin and Renée Gatignol
Laboratoire de Modélisation en Mécanique
Université Pierre et Marie Curie & CNRS
4 place Jussieu, 75252 Paris cedex, France
Abstract. Gas flow is considered on the basis of the discrete models for the Boltzmann equation. The formation and
propagation of disturbances in an initially uniform gas bounded by its plane-condensed phase in non-equilibrium with the
gas are investigated when the evaporation or condensation is taking place from the condensed phase. The obtained results
are in very good agreement with those given by continuum models of Boltzmann equation
INTRODUCTION
Let us consider a gas and its condensed phase located respectively in the half space y > 0 and y < 0 . Depending
on the conditions of the gas and of the condensed phase, evaporation or condensation takes place on the interface
y = 0 . Many authors on the basis of the usual kinetic theory have investigated this problem [1,2,3]. In this paper,
our purpose is to analyze it with simple discrete models and to pay attention to the formation and the propagation of
disturbances in the initially uniform gas bounded by its plane condensed phase in non-equilibrium with it. In discrete
kinetic theory the choice of the discrete velocity models is very important. First the models taken in consideration
are described. Then, the equations and the initial and boundary conditions are given. Finally, the numerical solutions
are obtained and described.
DESCRIPTION OF THE DISCRETE MODELS
In discrete theory, the integro-differential Boltzmann equation is replaced by a system of partial differential
equations more easy to analyze [4]. Here we use the two dimensional models related to the square [5] for a gas with
identical particles of mass m . In an orthonormal system of coordinates ( O x y ) , the reference velocity being c , the
r
r
r
velocities of a model are U i = u i i + vi j with − (2 p − 1) ≤ u i / c ≤ (2 p − 1) and − (2 p − 1) ≤ vi / c ≤ (2 p − 1) . We
point out that the quantities u i / c and vi / c have integer values and that p is an integer strictly larger than one.
(
(
)
)
r
r
r
U i = u i i + v i j with u i / c and v i / c integers positive or negative.
This model has 4 p 2 velocities. An example with p = 2 is shown on Figure 1. The summational invariants are
quantities attached to conservation laws [5]. Those attached to the physical conservation laws (mass, momentum,
energy) are called physical invariants. Discrete velocity models can possess other summational invariants called
spurious invariants. In that last case, the physical understanding of the results is confused. Fortunatly, the chosen
models have no spurious invariants if we take into account all the multiple collisions [6]. More, there are no
spurious invariants with the binary collisions only [5]. Here the dimension of the summational space is four.
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
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r
FIGURE 1. The model with 16 velocities (the velocity u i joint O to the number “i”).
r
As usual, we denote by N i the microscopic density of particles with the velocity U i . The macroscopic variables
r
(density n , velocity U and total energy E ) are:
n=
∑ Ni
i
r
r
r
, nU = ∑ N iU i , nE = ∑ N iU i2 .
(1)
i
i
r
The microscopic densities in a Maxwellian state associated with the macroscopic variables n , U and E are
r r
r
r
N i = exp α + β ⋅U i + γ U i2 where the parameters α , β and γ are unique and are determined by the implicit
relations (1) [4].
In this paper, we use the “kinetic” temperature T defined by the classical relation in the two-dimensional
r
physical space: kT / m + ( 1 / 2 ) U 2 = E where k is the Boltzmann constant. In discrete kinetic theory, this
temperature T is different from the “thermodynamic” temperature as that has been discussed by Cercignani [7].
(
)
FORMULATION OF THE PROBLEM
In the coordinate system O x y , the condensed phase occupies the region y < 0 and is at rest with a uniform
temperature T0 . The gas occupies the region y > 0 . The gas flow is uniform at infinity ( y = + ∞ ) with the density
n∞ , the temperature T∞ and the velocity ( 0 , v ∞ ) perpendicular to the interface separating the gas and its condensed
phase. At the time zero, the gas has a uniform flow with the macroscopic variables n∞ , ( 0 , v ∞ ) and T∞ . We
investigate the unsteady condensation ( v∞ < 0 ) or evaporation ( v∞ > 0 ) which takes place on the condensed phase
and the time development of the disturbance produced by the interaction of the uniform flow with the condensed
phase.
It is reasonable to search a solution depending on y and t only. The unsteady behavior of the gas is analyzed by
using the discrete velocity equations where only the binary collisions are taken into account. The set of the discrete
velocities is symmetrical with respect the interface y = 0 . So the boundary conditions of the diffuse reflection on
the interface are simple [8]. The molecules leaving the condensed phase constitute the corresponding part of the
Maxwellian distribution describing the saturated gas at the temperature T0 . These Maxwellian densities denoted by
N i 0 are associated with the temperature T0 , the velocity equal to zero and the macroscopic saturation density
denoted by n0 .
The system describing the evolution of the microscopic densities N i ( y ,t ) is:
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∂N i
∂N i 1
+ vi
=
∂t
∂y
2
∑ Aijkl (N k N l − N i N j ) ,
i∈Λ ,
y ∈ [0 , ∞ ]
(2)
j ,k ,l
r
r
r
r
where the coefficients Aijkl = S U i − U j α ijkl = S U k − U l α ijkl are the transition probabilities [4] with the cross
r r
r r
section S for binary collisions and the probability α ijkl for the collision ( U i ,U j ) → ( U k ,U l ) . We have introduced
the set Λ
of all the velocity indexes and we introduce the new sub-sets Λ+ = {i ∈ Λ , vi > 0} and
Λ− = {i ∈ Λ ,vi < 0} . There are no velocities parallel to the interface.
The boundary conditions on y = 0 are given only for the microscopic densities of the emerging particles that is:
N i ( 0, t ) = N i 0 , i ∈ Λ+ .
(3)
lim y →+∞ N i ( y ,t ) = N i∞ , i ∈ Λ ,
(4)
The conditions at infinity are:
where the densities N i∞ , i ∈ Λ , are the microscopic Maxwellian densities associated with the macroscopic density
n∞ , the total energy E ∞ = kT∞ / m + ( 1 / 2 ) v∞2 and the velocity ( 0 , v ∞ ). At last the initial conditions in t = 0 are:
N i ( y ,0 ) = N i∞ , i ∈ Λ .
(5)
BASIC EQUATIONS AND NUMERICAL APPROACH
All the quantities are taken dimensionless with the reference quantities: c for the velocities, n0 for all the
densities, λ0 for the length and λ 0 / c for the time, where λ 0 = 1 / S n 0 is the mean free path of the saturated gas at
the temperature T0 . We introduce the following notations:
ni = N i / n0 , ni 0 = N i 0 / n0 , ni∞ = N i∞ / n0 , ω i = vi / c , ∀i ∈ Λ ,
r r
ν = n / n0 , ν ∞ = n∞ / n0 , v = U / c , ω ∞ = v∞ / c , η = y / λ0 , τ = Sn0 ct ,
(6)
θ = 2kT /( mc 2 ) , θ 0 = 2kT0 /( mc 2 ) , θ ∞ = 2kT∞ /( mc 2 ) , aijkl = Aijkl / Sc .
Finally the problem that we have to solve in the half space η > 0 is given here after:
∂ni
∂n
1
+ωi i =
∂τ
∂η 2
∑ aijkl ( n k nl − ni n j ) ,
i ∈ Λ , η ∈ [0 , ∞ ] ,
j ,k ,l ,
ni ( η ,0 ) = ni∞ , i ∈ Λ ,
(7)
+
ni ( 0,τ ) = ni 0 , i ∈ Λ ,
limη →+∞ ni ( η ,τ ) = ni∞ , i ∈ Λ .
It is easy to see that the microscopic densities ni 0 and ni∞ depend on the four quantities ν ∞ ,θ 0 ,θ ∞ , ω ∞ . In
other words, the macroscopic non-dimensional parameters, which control the problem, are dimensionless density
ν ∞ of the gas at infinity, the dimensionless temperatures θ 0 and θ ∞ of the condensed phase and of the gas at
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infinity and the dimensionless velocity ω ∞ of the gas at infinity. We have four independent parameters. In the
continuum kinetic theory, the same problem is characterized by three parameters [1,2,3]. This difference can be
explained by the fact that in the discrete kinetic theory it exists a reference velocity related to the model (for
example the smaller velocity of the discrete model).
We analyze the unsteady problem (7) by a splitting method [9,5]. In that actual computation, we consider the
problem in a finite region 0 ≤ η ≤ D and impose in η = D a part of the conditions at infinity, i.e.:
ni ( η , D ) = ni∞ , i ∈ Λ− .
We pay attention that we have reduced the conditions at infinity only to the densities ni with i ∈ Λ− , that is for the
particles, which come into the computation domain. The spatial derivatives and the collision terms are treated
separately as it is shown hereunder:
nim +1 / 2 − n im 1
=
2
h
∑ aijkl ( n km nlm − nim n mj )
with
ni0 = ni∞ , ∀i ∈ Λ ,
(8)
j ,k ,l
nim +1 − nim+1 / 2
dn m +1
+ ωi i
=0
h
dη
∀i ∈ Λ ,
(9)
with:
nim +1 ( 0 ) = ni 0 , ∀i ∈ Λ+
and
nim +1 = ni∞ , ∀i ∈ Λ− ,
where the time step is h , where nim is the density at time τ = mh ( m = 0 ,1,... ) and nim+1/ 2 the density at the
intermediate time. The quantities nim +1 / 2 are depending on η . A numerical explicit method is used to solve the
differential system (9) [5].
NUMERICAL RESULTS
The initial and boundary value problem has been numerically solved for the two cases of the condensation and
evaporation. The discontinuity between the boundary condition at η = 0 and the initial data generates a disturbance
propagating into the gas. We are interested in the long time evolution of the disturbance.
Like in the previous works using the continuum Boltzmann equation [1,2,3,10], we analyze the results obtained
for a large number of initial or boundary conditions. The solutions are classified into different types [5]. In this
paper we give the results with some details for the condensation case and we briefly summarize those corresponding
to the evaporation case.
Condensation Problem
Some typical examples of the different types of solutions are shown on Figures 2 and 3 with the graphs for the
density ν or the temperature θ only. For the large times, we observe several behaviors: The solutions approach the
solution of the steady problem or the solutions develop a wave, which propagates to infinity.
On Figure 2, the density ν is decreasing when η increases. The gas is compressed on the condensed phase and
the compressed region propagates toward infinity. The speed of propagation slows down and tends to zero. The
solution ( ν ,θ ) approaches the steady state ( ν ∞ ,θ ∞ ) associated with the given conditions at infinity. The steady
state is approached very rapidly. For the parameter values corresponding to Figure 2, it is attained about for τ = 10
(i. e. for t equal to 10 mean free times). We have not a disturbance. This solution is called "type I" [11].
On Figure 3-a, the gas is compressed on the condensed phase, and a compression wave propagates to infinity.
Through this wave, when η increases, the density ν decreases from a constant value to the value ν ∞ . With the
parameter values corresponding to Figure 3, the thickness of this compression wave is about 10 in the variable η
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i.e. about 10 mean free paths. It is also possible to evaluate its speed of propagation: it is about 0.1 (that is 0.1 c ).
The region behind the wave approaches a steady state with a new Maxwellian state. When the time tends to infinity,
this Maxwellian state tends to occupy the entire gas region. This solution is called of "type II" [11].
We have also the solution of "type III" [11] where a rarefaction region develops on the condensed phase and
diffuses as time goes on. When the time tends to infinity the steady state with the given conditions at infinity is
obtained [11].
In the same way, we have also a rarefaction region developing on the condensed phase and an expansion wave
propagating to infinity. As for the compression case (solution of "type II") the region behind the wave approaches a
steady state with a Maxwellian state different of the given state at infinity. This solution is called "Type IV" [11].
For the temperature, an example of a such solution is given on Figure 3-b.
Other examples of these solutions ("types I to IV") are given in the thesis of the first author [5]. Of course, like
on Figures 2 and 3, we observe the Knudsen layer closed to the condensed phase.
As previously said, the condensation problem depends on four parameters ( ν ∞ , θ 0 , θ ∞ , ω ∞ ). We have fixed
two of them θ 0 / θ ∞ and ω ∞ . Then according to the types of solutions that we have, the two dimensional space
( ν ∞ , ω ∞ ) is divided into two or four regions. With θ ∞ / θ 0 =1, if ω ∞ is small ( ω ∞ < 1.5 ), we find two regions
corresponding to the solutions of type II and IV (Figure 4-a) and if ω ∞ is bigger than 1.5, we find four regions
corresponding to the four types of solutions (Figure 4-b). On that last figure, we observe that the four regions are
intersected in a point ( ν c ,θ c ) corresponding to a steady solution.
Evaporation Problem
In that case, by the analysis of the results obtained for a large number of the initial and boundary data (with
ω ∞ > 0 ), the solutions of the unsteady problem are classified in four types. These solutions are different of the
solutions of the condensation problem. We can observe solutions, which contain a compression wave (Figure 5-a), a
rarefaction wave (Figure 5-b) or two waves at the same time (Figures 5-c and (5-d). In these descriptions the waves
are propagating away from the condensed phase. Uniform regions separate the waves. A detailed discussion is given
in the first author thesis [5]. In this work, some attention is also paid to the existence or not of the steady solutions in
the two cases of the condensation and of the evaporation.
CONCLUSION
All the previous results are obtained by using the simplest model with 16 velocities (i.e. p = 2 ). The same studies by using
models with a greater number of velocities have been made ( p = 3 , 4 or 5). The so-obtained results are very closed. The
differences are canceled completely between models with p = 4 and p = 5 [5].
The results obtained in the basis of the discrete kinetic theory are in very good agreement with the results of Sone, Aoki and
their collaborators [2,3,10,11]. The parameters retained in their papers are the ratios p ∞ / p 0 , T∞ / T0 and the Mach
number at infinity. In our analysis, we use four parameters (and not three) and the quantitative comparison is a little
difficult.
But qualitatively, for the condensation case, the present results are in agreement with whose of Aoki and al. [4].
The same four types of solutions are founded and the regions for the different types of solutions have the same form.
For evaporation case, for a ratio of temperatures equal to one, the same types of unsteady solutions are obtained. In
the space ν ∞ , ω ∞ the four regions for the four types of solutions have also the same form [5].
In conclusion, we emphasize that the results here obtained with very simple models (with a very small number of
velocities) are in a very good agreement with the results obtained by using the continuum Boltzmann equation. Now,
it will be interesting to establish the mathematical prove of these results. The simplicity of the model with 16
velocities can help.
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FIGURE 2. The evolution of the quantities ν and θ / θ 0 in function of η for θ ∞ = θ 0 = 2 ,
ν ∞ = 12 and ω ∞ = −2 and using the model with 16 velocities.
(3-a)
(3-b)
FIGURE 3. The evolution of the quantities ν and θ / θ 0 in function of η for θ ∞ = θ 0 = 2 , ω ∞ = −1 ,
and ν ∞ = 4 on (3-a) and ν ∞ = 4.9 on (3-b), using the model with 16 velocities.
(4-a)
(4-b)
FIGURE 4. The regions in the plane ( θ ∞ , ν ∞ ) with θ 0 / θ ∞ = 1 and ω ∞ = −1 (4-a) or ω ∞ = −2 (4-b).
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(5-a)
(5-b)
(5-c)
(5-d)
FIGURE 5. Different density profiles in the case of the evaporation.
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2.
3.
4.
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