Evaporation and Condensation from or onto the
Condensed Phase with an Internal Structure
Y. Onishi and K. Yamada
Department of Applied Mathematics and Physics, Tottori University, Tottori 680-8552, Japan
Abstract. Motions of a vapor due to evaporation and condensation processes from or onto the plane condensed phase with a
temperature field as its internal structure have been studied based on the Boltzmann equation of BGK type. The condition of
the continuity of energy flow across the interface has been derived and used in the formulation of the problem at the kinetic
level. Moreover, the interface surface is considered to have some kind of imperfectness, which may be expressed in terms of
a parameter (called the imperfectness parameter here) first introduced by Wortberg and his colleague. The results obtained
describe appropriately the development of the transient flow fields due to the processes of evaporation and condensation
caused by the continuous energy supply from or onto the condensed phase. Some of the new features are that the role of Γ, a
parameter associated with the latent heat of vaporization, is totally different from the case in which the condensed phase has
no internal structure; The imperfectness parameter coupled with the effects of the internal structure of the condensed phase
affects the contact region but not the shock wave produced. Moreover, the insight of the formation of a shock wave can be
grasped by looking at its early stages, which are associated with the series of weak compression waves catching up with one
after another.
INTRODUCTION
Flow problems associated with phase-change processes are not only of theoretical interest but also of practical importance because they do have wide applications in various thermo-fluid engineering fields. Such problems, however, are
the ones to which the ordinary continuum-based fluid dynamics is not directly applicable because of the existence of
a nonequilibrium region called the Knudsen layer in the close vicinity of the condensed phase. This region can never
be neglected in any problems of such kind even in the continuum limit. Their analyses, therefore, must necessarily
be based on the kinetic equation because it is this nonequilibrium region that is responsible for the phase-change
processes to occur. For this reason, the appropriate analysis of the problems, regardless of whether it is theoretical or
numerical, imposes us a great burden. In spite of this, a large number of evaporation and condensation problems have
been studied so far based on the kinetic equation and a fairly large amount of the useful information and knowledge
has been obtained. However, the analyses so far seem to have been confined to systems in which the condensed phases
have no internal structures except probably a work by Sone and Aoki [1] who treated the slow motions of a volatile
particle based on the linearized asymptotic theory developed in general terms by Sone and Onishi[2]. The consideration of the internal structure within the condensed phase corresponds to practically important flow problems within
heat and energy exchange or transfer systems such as heat pipes. At the same time, it may also serve to clarify a certain
aspect of the characteristics of the imperfectness of the interface of the condensed phase. Therefore, we here try to
study the effects of the internal structure of the condensed phase on the flow fields and their transition processes to the
final (steady) states by taking up a simple flow field of a vapor due to evaporation and condensation processes from
or onto its plane condensed phase with a temperature field as its internal structure. In addition, some imperfectness of
the interface surface of the condensed phase is also taken into account in the present study, which may be expressed
in terms of a parameter α c (called imperfectness parameter here) first introduced by Wortberg and his co-workes [3].
For this analysis, the equation of heat conduction with constant substance properties is used for the temperature field
within the condensed phase, and for the motions of the gas phase the Boltzmann equation of BGK type [4] is used
subject to the diffusive type of boundary conditions together with the condition of the continuity of energy flow across
the interface. The results will reveal some of the new aspects of the present problem. Among others, 1) the role of Γ,
a parameter associated with the latent heat of vaporization, is totally different from the case in which the condensed
phase has no internal structure; 2) the imperfectness parameter α c coupled with the existence of the internal structure
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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of the condensed phase does not affect so much the pressure and velocity fields but does the density and temperature
fields, indicating that it affects the contact region but not the shock wave produced; 3) the formation process of a shock
wave can be grasped clearly by looking at its early stages of development associated with the series of compression
waves catching up with one after another.
FORMULATION OF THE PROBLEM
Consider a two phase system of a vapor and its condensed phase with finite thermal conductivity. Let the interface
surface of the condensed phase be at x 0 and the other end surface at x D. The vapor occupies the half-space
( x 0 ). Initially, the condensed phase and the vapor phase are in complete equilibrium at a temperature T 0 . Let the
pressure and density (or number density) of the vapor at this state be P 0 and ρ0 (or N0 ), respectively. Suppose that
at a certain time, say t 0, the temperature of the edge surface of the condensed phase at x D, which is initially
T0 , is suddenly changed to Tc . The heat flow then occurs through the condensed phase and after a certain elapse of
time the temperature of the interface surface starts to be changing with time. This leads to the onset of phase change
processes at the interface of the condensed phase, giving then rise to transient motions of the vapor. This is what the
present study is concerned with.
Now, for the temperature field within the condensed phase, the equation of heat conduction with constant substance
properties
∂T̃
∂2 T̃
κ 2 0
(1)
∂t
∂x
holds for D x 0 , where t is the time; x is the coordinate; κ is the thermal diffusivity of the condensed phase,
which is here assumed to be constant for simplicity, and T̃ represents the temperature field within it. For the description
of the motions of the vapor in the present problem, on the other hand, the Boltzmann equation of BGK type [4] is used,
which may be written as
∂f
∂t
ξx ∂∂xf Nνc Fe f with
Fe (2)
ξx u2 ξ2y ξ2z N
exp
2RT
2πRT 32
N
Nu
3
2N kT
P N kT
1
ξx
1
2
2
2
2 mξx u ξy ξz (3)
f dξx dξy dξz ρ R T
(4)
(5)
where ( ξx ξy ξz ) is the molecular velocity vector; f is the molecular velocity distribution function, F e being the
local Maxwellian distribution characterized by the local fluid dynamic quantities; N, u, T , P and ρ are, respectively,
the number density, the velocity, the temperature, the pressure and the density of the gas; m is the molecular mass;
k is the Boltzmann constant and R km the gas constant per unit mass of gas. ν c is a constant associated with the
collision frequency ( Nνc is the local collision frequency) and, hence, can be calculated either from the viscosity µ
or from the thermal conductivity λ of the gas at a certain reference state, say, at the initial equilibrium state, by the
following relation
P0
5 P0
R (6)
N0 νc µ0
2 λ0
the suffix 0 being understood to indicate the quantities at the initial state. It may be noted here that, in the BGK model
equation, the relation λ 52R µ holds and, hence, the Prandtl number is unity for this model equation.
The initial condition for the distribution function f for the present problem is, at t
f
ξ2x ξ2y ξ2z N0
2RT
2πRT
exp
32
0
0
631
for x 0
0
(7)
for all possible values of the components of the molecular velocity vector. As to the boundary conditions, the condition
for f at infinity (x ∞) is exactly of the same form as the initial condition (7) for all times and for ξ x 0. The
condition for the distribution function f at the interface surface of the condensed phase (x 0), however, which is the
specification of the distribution function for molecules leaving the surface of the condensed phase ( ξ x 0 ), may be
specified as follows:
f
ÑW
2πRT
32 exp W
with
ÑW
ξ2x ξ2y ξ2z 2RTW
αc NW 1 αcND
for ξx 0
at x 0
0 αc 1
(8)
(9)
where TW represents the temperature of the interface surface of the condensed phase, which is yet an unknown
parameter, and will eventually be determined as part of the solution. ÑW is the number density for those molecules
emitted from the surface. α c is a parameter associated with some kind of the imperfectness of the interface surface
first used by Wortberg and his co-workes [3] in their experimental and approximate analytical studies on this kind
of problems, and here it is called the imperfectness parameter. The meaning of the parameter α c may be clear.
It represents that a cetrain fraction 1 α c of the molecules leaving the condensed phase is subject to a velocity
distribution of diffusive type holding at a solid boundary, namely, a Maxwellian distribution function characterized
by the temperature and velocity of the condensed phase but with the number density being determined by the no net
mass flow condition across the boundary surface. The quantity N D represents that number density of the molecules
and, therefore, is obtained from the distribution function for incident molecules to the condensed phase (ξ x 0) as
2π
(10)
ND 2πRTW 12 ξx 0 ξx f dξxdξydξz From this, it may be clear that ND is also yet an unknown constant to be determined as part of the solution. The
quantitiy NW , on the other hand, is a unique function of the temperature T W of the surface of the condensed phase
and here is taken as the saturated vapor number density corresponding to T W . NW , therefore, is calculated from the
Clapeyron-Clausius relation (see e.g., Landau & Lifshitz [5]) as
NW
kPTW W
PW
P0
exp Γ
T0
TW
1
Γ
hL
R T0
(11)
where hL is the latent heat of vaporization per unit mass and Γ is a nondimensional parameter associated with the
latent heat. In addition to the above specification of the distribution function at the interface surface of the condensed
phase, the condition of the continuity of energy flow across the interface surface has to be imposed, which may be
given as
λc ∂∂xT̃ H ρ uhL ρ uh at x 0
(12)
where λc , assumed constant, is the thermal conductivity of the condensed phase itself. h is the enthalpy per unit mass
of the gas and h c p T , c p being the specific heat at constant pressure. H is the energy flux of the gas, which is here
evaluated at x 0, and is given by
1
m
ξx ξ2x ξ2y ξ2z f dξx dξy dξz
(13)
H 2
1π ÑW k TW 2RTW 12 12 m
ξx ξ2x ξ2y ξ2z f dξx dξy dξz (14)
ξx 0
where the condition (8) for the distribution function f has been taken into account in the above expression.
Moreover, the initial and boundary conditions for the temperature T̃ of the condensed phase should be specified
and these are:
T̃ T0
for D x 0 and t 0
Tc
T̃ TW
T̃
at
at
x D
x0
for
for
Note that the temperature of the interface surface TW varies with time.
632
0
t 0
t
(15)
TABLE 1.
Values of the parameters for various substances
Substances
T0 Æ K
λλ0
κκ0
Γ hL RT0 hL JKg
R JKgÆ K
H2 O
370
28
0.0074
13
2265325
454.818
CO2
216.58
15
0.0926
9
345445
170.327
CO2
255
10
0.3263
7
278460
148.759
Argon
90
21
0.0607
9
159057
199.522
CHARACTERISTIC PARAMETERS
For the analysis of the present problem, we have introduced the length scale L and the time scale τ 0 taken as
L
τ0 π
2
l0 12
L
2RT0 12
2
γ
µ0
c0
P0
γ 12
2
L
c0
1
N0 νc
µ0
P0
with
l0 µ0
P0
8RT0
π
12
(16)
where c0 is the sound speed of the gas at the initial state defined by c 0 γRT0 12 , γ being the specific heat ratio
( γ 53 here). l 0 is the mean free path of the gas molecules at the initial state. It may be mentioned that, since the
length scale L is of the order of the molecular mean free path l 0 , the time scale τ0 adopted here represents the mean
collision time of gas molecules at the uniform initial state. With the fluid dynamic quantities at the initial state together
with these length and time scales, the system of the governing heat conduction and kinetic equations and the initial
and boundary conditions is appropriately nondimensionalized, giving the following non-dimensional parameters
λc
λ0
κ
κ0
Tc
T0
Γ
αc D
L
(17)
characterizing the present problem, where κ 0 is the thermal diffusivity of the gas at the initial state defined by
κ0 λ0 c p ρ0 . The specification of these parameters give the flow field as the functions of xL and t τ 0 . Incidentally,
a list of the values of the sample substance parameters is shown in Table 1 in order to have rough ideas about the values
of the parameters.
RESULTS
With a difference scheme applied, the system of the governing equations has been solved numerically subject to the
initial and boundary conditions specified. First of all, some of the results of the development of the transient flow
fields associated with the evaporation process are shown in Figs. 1 and 2, where the shock waves produced due to the
evaporation processes followed by the contact regions are clearly visible. The conditions for both cases are the same
except for the value of the imperfectness parameter α c , i.e., Fig. 1 for α c 10 and Fig. 2 for α c 075. The two
flow fields are almost the same except the density and temperature fields where slight differences between the two
can be noticed in the region bewteen the interface surface and the contact region. The pressure and velocity fields are
not so much affected by the imperfectness parameter α c . Moreover, the Mach numbers of the shock waves produced
are almost the same. This might indicate that α c coupled with the effects of the internal structure of the condensed
phase (the finite thermal conductivity effects) affects the contact region but not the shock wave produced. When the
condensed phase has no internal structure (infinite thermal conductivity case), however, the effect of α c was to weaken
the evaporation processes and, hence, the shock waves produced (see Onishi et al. [6]). As to the transient processes, it
seems that the flow fields almost establish themselves at about time t τ 0 2000. However, the shock waves still have a
rounded part at their rear edges, which means that a series of the compression waves, which are continuously catching
up with the shock waves, are on the way to merge and strengthen the shock waves. These compression waves are those
produced by the evaporation processes, whose strengths are still changing in time owing to the changes in temperature
TW of the interface surface. The rounded part of the shock waves will eventually disappear as the flow fields approach
their final states to be established. The shock waves, then, have fairly sharp edges as they should. Unfortunately, in
633
these figures, the Knudsen layers are not visible because of the scales of the x -axis of the figures but the layers are
surely formed in the close vicinity of the interface surfaces of the condensed phase. The steep gradients of the fluid
dynamic quantities exist within these layers and for this indication the values of the fluid dynamic quantities at the
interface surface are listed in the captions of the figures.
ρ
ρ0
P
P0
1.3
1.3
1.2
1.2
4000
1.1
2000
1000
3000
4000
3000
1.1
2000
1000
1.0
1.0
0
2000
x/L
4000
0
2000
x/L
4000
u
c0
1.1
T
T0
4000
0.1
3000
2000
1000
1000
1.0
2000
3000
4000
0.0
0
2000
x/L
4000
0
2000
x/L
4000
FIGURE 1. Transient distributions of the fluid dynamic quantities (evaporation case). λc λ0 300, κκ0 032, Tc T0 20,
Γ 110, αc 10 and DL 200. The Mach number of the shock wave is about Ms 1079 (at about t τ0 4000). The numbers
in the figures indicate the time t τ0 . Note that ρρ0 x0 13534, PP0 x0 13467, T T0 x0 09951 and uc0 x0 01313
at t τ0 4000.
The jumps in temperature at the interface surface, which are the cause of the phase change processes and give a
measure of their strength, are shown in Fig. 3 for different sets of the values of Γ, λ c λ0 and κκ0 . These jumps
become large as the value of Γ becomes small, which means that the phase change processes at the interface and,
therefore, the shock waves produced strengthen as Γ decreases. This can be understood from (the non-dimensional
form of) Eq. (12). Since the term λ c ∂T̃ ∂x balances mainly with the term ρ u h L , the small temperature gradient
∂T̃ ∂x within the condensed phase is sufficient for small values of h L (or Γ in its non-dimensional form) for a given
value of λc λ0 , which means that the temperature TW of the interface surface can become large compared with T 0 .
This leads to the fairly large temperature jumps TW T 0TW and, hence, the pressure jumps PW P0PW
at the interface, which is the cause of the strong phase change processes compared with the case of relatively large
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ρ
ρ0
P
P0
1.3
1.3
1.2
1.2
4000
1.1
2000
1000
3000
4000
3000
1.1
2000
1000
1.0
1.0
0
2000
x/L
4000
0
2000
x/L
4000
u
c0
1.1
T
T0
4000
0.1
3000
2000
1000
1000
1.0
2000
3000
4000
0.0
0
2000
x/L
4000
0
2000
x/L
4000
FIGURE 2. Transient distributions of the fluid dynamic quantities (evaporation case). λc λ0 300, κκ0 032, Tc T0 20,
Γ 110, αc 072 and DL 200. The Mach number of the shock wave is about Ms 1079 (at about t τ0 4000).
The numbers in the figures indicate the time t τ0 . Note that ρρ0 x0 13353, PP0 x0 13456, T T0 x0 10077 and
uc0 x0 01313 at t τ0 4000.
values of Γ adopted here. 1 This feature forms a striking contrast to the case in which the condensed phase has no
internal structure (infinite thermal conductivity case). As to the role of the parameter λ c λ0 , when the value of λ c λ0
is large, the temperature gradient ∂ T̃ ∂x within the condensed phase can become small for a given value of Γ. This
means the existence of large temperature jumps at the interface and hence the strong phase change processes occurring.
Naturally, from this, the phase change processes at the interface are the strongest when the condensed phase has no
internal structure which corresponds to its thermal conductivity being infinite (λ c λ0 ∞).
Finally, one of the results for the transient flows due to the condensation processes is shown in Fig. 4, where it is
seen that the expansion waves are propagating in the flow field. As time goes on, the uniform region is being formed
behind the expansion waves. Between the edge of this uniform region and the interface surface of the condensed phase,
It should be noted, however, that the very large values of Γ gives extremely large values of PW P0 for a given value of TW T0 and, hence, the
very strong phase change processes.
1
635
4000
T
T0
(a)
1000
1.1
(b)
T
T0
4000
1.1 1000
500
500
4000
1000
200
500
1.0
1.2
T
T0
500
4000
200
1.0
200
−2
500
200
1000
0
2
4000
1000
(c)
x/L 4
−2
0
1.2
2
x/L 4
(d)
T
T0
4000
1000
500
1.1
1.1
4000
200
200
500
0
500
200
200
1.0
−2
4000
1000
1000
1.0
2
x/L 4
−2
0
2
x/L 4
FIGURE 3. Transient temperature jumps occurring at the interface at x 0 (evaporation case). Tc T0 20, αc 10 and
DL 200 ( 20 xL 0 : the condensed phase). The numbers in the figures indicate the time t τ0 . a: Γ 70, λc λ0 300,
κκ0 032. b: Γ 110, λc λ0 300, κκ0 032. c: Γ 70, λc λ0 500, κκ0 05. d: Γ 110, λc λ0 500,
κκ0 05.
there is formed the Knudsen layer, where the steep gradients of the fluid dynamic quantities exist. Although the layer
is not so clearly visible in the figure, the values of the fluid dynamic quantities at the interface surface, which are listed
in the caption of the figure, would indicate the existence of the layer and the magnitudes of the gradients of the fluid
dynamic quantities themselves.
ACKNOWLEDGMENTS
This work was partially supported by the Grant-in-Aid for Scientific Research (No.14550153) from the Japan Society
for the Promotion of Science.
636
1.0
1.0
ρ
ρ0
P
P0
4000
3000
2000
4000
3000
1000
2000
0.9
0.9
1000
0
2000
4000
x/L
T
T0
0.8
0
2000
4000
x/L
0.0
1.00
u
c0
4000
4000
3000
2000
3000
0.98
1000
2000
1000
0.96
−0.1
0.94
0
2000
x/L
4000
0
2000
x/L
4000
FIGURE 4. Transient distributions of the fluid dynamic quantities (condensation case). λc λ0 300, κκ0 032, Tc T0 05,
Γ 110, αc 10 and DL 200 ( 20 xL 0 : the condensed phase). The numbers in the figures indicate the time t τ0 .
Note that ρρ0 x0 08223, PP0 x0 08181, T T0 x0 09949 and uc0 x0 01074 at t τ0 4000.
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