Nonequilibrium Processes at Evaporating and Condensing Surfaces Aleksei K. Rebrov

Nonequilibrium Processes at Evaporating and Condensing
Aleksei K. Rebrov
Institute ofThermophysics, Novosibirsk, Russia
E-mail: [email protected]
Abstract. Nonequilibrium processes in Knudsen layers in the vicinity of evaporating and condensing surfaces are
considered. Results of the previous publications and some new data were used in the presented analysis.
The problems of evaporation-condensation were the subject of systematic inquiry during the last forty years.
Well-known and explicable interest to these problems was combined with an aspiration for solving the fundamental
problems of formation of flows with a discontinuity at boundaries that are phase transition interfaces. Beginning
from the early works [1-4] up to that appeared during the last years [5, 6], the vast scope of knowledge was
accumulated and a number of problems on flow formation and structure was solved. In spite of using for analysis the
simplest hard sphere molecular model in most cases, general and important conclusions are stated. It is worth to note
here the determination of feasible flow regimes by evaporation-condensation [7-9] and quantitative data on
macroscopic boundary conditions (parameters jumps) on phase transition interfaces [5, 10-12]. One can restore the
history of studies on references of above cited papers.
The aim of the present work is analysis of nonequilibrium processes in the Knudsen layer at condensing and
evaporating surfaces. The analysis of these processes in the form of the presentation of the distribution function of
molecular velocities (DFMV) or DFMV superposition is hampered by surveying of data with the information
surplus. That is why the DFMV convolution in the form of temperature components on directions was chosen as the
characteristic of nonequilibrium phenomena. The results of direct simulation by the Monte Carlo (DSMC) method of
evaporation-condensation processes at the surfaces of canonical forms (flat surfaces, cylinder and sphere) were
widely enlisted.
The most detailed information on results of the computational solution of the Boltzmann equation in BhatnagerGross-Krook model for the case of the evaporation-condensation transfer is presented in [13]. The typical example
of these results is shown in Fig. 1, where the relative pressure PIPa and temperature T/Ta distributions between
surfaces are given for the regime with the Knudsen number in the range of oo > Kn > 0.01, Tf/Ta =1.1; P\/Pa = 3.
The surface b is located to the right. Kn = k/d = J&RTa/n /dva. Here A- is the mean free path of molecules, d is the
gap between the surfaces, va is the collision frequency. It should be noted that the flow is subsonic one and so called
inverted temperature gradient is localized by the condensing surface at Kn < 0.01. The occurence of the inverted
temperature gradient, directed to the colder condensing surface, was discovered first in the theoretical work [14] and
up to now is not confirmed by reliable experiments. This phenomenon was intriguing during about twenty years and
gave birth to more than tens of publications cited in [13, 15 - 17].
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
The inverted temperature gradient which is impossible by the fully condensing surface takes place at certain ratio
of the vapor flux to be condensed and the contrary molecular vapor flux from the surface corresponding to the
saturation pressure of vapor. This effect can be qualitatively explained on the basis of evaluations, using the
comparison of terms of the Navier-Stokes equations for the flow and boundary jumps of density and temperature
1 OS-
FIGURE 2. a, c — Kn = 0.01, 1; b, d — Kn = 0.1; curves 1 - 4 correspond to Jj|, T±, T and the Mach number,
respectively. R = (r- r0)A; r0 is the sphere radius, A is the molecular mean free path at infinity.
The essence of this phenomenon is explicitly exposed by DSMC modeling of the flow near the condensing
surface. The structure of the flow in the case of condensation on the "black" or fully absorbing sphere and the sphere
with finite saturation pressure was studied in [18], Fig. 2 shows the radial distribution of the Mach number, the
parallel Ty (along the flow), perpendicular T± (across the flow) and average T = (Ty + 2T±)/3 temperatures,
normalized by the stagnation temperature.
The cases a, b, c correspond to the Knudsen numbers equal to 0.01; 0.1; 1 for the black sphere, and case d
corresponds to Kn = 0.1 and (Poo - Pw)/Pao = 0.5. Here Pw and POD are saturation pressure and pressure at infinity. The
perpendicular temperature in the Knudsen layer is noticeably higher than average and parallel ones. The increase of
the average temperature downstream the flow at some distance from the surface (the inverted temperature gradient)
is pronounced at Kn ~ 1. At the finite saturation pressure and Kn = I (case d) this effect becomes considerable. In
both cases it happens in the zone with the temperature anisotropy. The only explanation of the inverted temperature
for the case c is the formation of the elevated temperature of the shock type in consequence of the collision of the
collapsing molecular flows at high Knudsen numbers. In the case of Kn = 0.1 this effect is forced by flow of
evaporating molecules.
0.04 x/d
0.008 x/d
FIGURE 3. Pb/Pa = 2; a — the Knudsen layer is not discernible; b — temperature anisotropy at the surface.
The effect discovered in [18] is definitely pronounced in the nonequilibrium zone (with the temperature
anisotropy). The data of [13] for flat surfaces are obtained for the region of Kn > 0.01. The interest arises to compare
the extents of Knudsen layer zones and zones of localized inverted temperature gradient for flat parallel flow.
Author and M. Yu. Plotnikov carried out this comparison using the DSMC modeling of the overcondensation
between two parallel surfaces at P*/Pa = 2 and Tj/Ta = 1.05, 1.12, 1.13, 1.15, 1.5 for Kna = 0.01. Kna is determined
on the molecular mean free path at parameters Ta and Pfl. Fig. 3 shows the data for iyrfl = 1.05 and 1.5: in Fig.3, a
and 3, b the distance between the surfaces is about five and one molecular mean free path, respectively. The
obtained average temperatures far from the surfaces coincide with the data presented in [13]. The data in Fig. 3b
close to the flat surface can be considered as qualitative ones. Nevertheless one can state the almost total absence of
the Knudsen layer. So the inverted temperature gradient in the case of TiJTa = 1.05 stretches overall temperature
boundary layer. The evolution of the temperature distributions in the regimes of transition to the inverted
temperature gradient is shown in Fig. 4. The Knudsen layer close to the point of temperature gradient inversion takes
not more than two mean free paths. One can observe the origin of the inverted gradient by changing TiJTa from 1.12
to 1.15. The perpendicular temperature here unlike the case illustrated by Fig. 2 is lower then parallel one. At Ta =
const by changing Tj/Ta from 1.15 to 1.12 the temperature gradient out of the Knudsen layer becomes inverted. In
the Knudsen layer the average temperature just close to the surface has tendency to decrease that is not sensed by the
computational solution of the Boltzmann equation [13],
One can state the smooth transition of temperature gradient from positive to negative. It is clear that at constant
Pa with the decreasing TiJTa the relative disturbance in the Knudsen layer, produced by encountering flows,
increases due to decreasing the mass flux with the reduced temperature in the undisturbed region, and the inverted
temperature gradient appears.
In the case of the overcondensation between flat surfaces at Kn —> 0 the vapor velocity can not exceed the sonic
one. However, at Kn of the order of 0.01 and high value of P&/Pa the flow velocity becomes supersonic far from the
condensing surface, in the region of the continuum flow with the local maxwellian DFMV. One can find the
illustration of this fact in [13] in the case Kn = 0.01, Pb/Pa = 100 and Tb/Ta = 2. It should be noted that in this
situation the transition through the sonic velocity takes place at the uniform geometrical effect in consequence of the
viscous forces action. One can state that such transition can be possible only at the specific boundary conditions.
We'll touch upon this problem below.
T 1
FIGURE 4. a,b,c — TiJTa =1.15,1.13, 1.12, respectively; P^Pa = 2.
The steady expansion of the evaporated gas from cylindrical and spherical surfaces into low density background
or vacuum is thoroughly investigated in [19, 20] in the frames of the Boltzmann equation solution using the
approximations of Boltzmann-Krook-Welander and Bhatnager-Gross-Krook equations. The radial distribution of the
macroscopic parameters is given in details for cylindrical source at Kn > 0.01 and for spherical one at Kn > 0.005.
For both cases the asymptotic solutions at Kn —» 0 are obtained. For some cases the DFMV evolution is illustrated.
One of peculiarities of the evaporation from the surface of radial sources in the low-density background is the
occurrence of strong gradients of parameters. It may cause the essential nonequilibrium in expanding flows. At Kn >
0.01 the nonequilibrium in radial flow and in the Knudsen layer of the evaporating gas can be overlapped. The radial
distribution of parameters in such cases can be of interest in particular cases. To elaborate the general view on
formation of radial vapor flows it is necessary to analyze the regimes with separated zones of nonequilibrium.
In our studies of radial flows the well-known DSMC method [21] was used. An important point is that when
using this method no boundary condition problems arise. For the flow from cylindrical source [22] the relative
departure of the radial component of temperature from the average one (T- Tr)/Tin the transonic zone (M ~ 1) was
calculated in dependence on the Knudsen number in the region at Kn > KT4 (Fig. 5). It immediately follows that at
these conditions the entire supersonic region is nonequilibrium. The results of detailed calculations have shown that
at Kn > 10~3 the nonequilibrium zones of expansion and the Knudsen layer are merged. Only for the case Kn < 10"4
one can talk about continuum flow in the transonic zone.
While the Knudsen number decreases the flow in the vicinity of a surface tends to the flow close to the flat
surface. It is characterized by the relative fraction of molecules ijie returning back after evaporation according to the
Hertz-Knudsen law and following collisions (Fig. 6). This value is essentially independent of the condensation
coefficient a, accepted in the used model only for returning molecules. At Kn = 10~4 the value ijie approximates to
its limit that is in agreement with data of other works [23, 24],
The radial location of the sonic surface (M = 1) as a number of mean free paths (r - r0)A, depending on the
Knudsen number is shown in Fig. 7. The obvious conclusion follows from this figure: at Kn < 10"3 the sonic surface
is located in the continuum flow with the local maxwellian DFMV: one can call this zone as <sNavier-Stokes region".
In this important statement it is supposed that the Knudsen layer confines itself within the limits not more than 10
mean free paths. It follows both from our calculations and the data of other works for relatively weak geometrical
effect on the length of the molecular mean free path.
The behavior of the macroscopic parameters in the subsonic zone of the flow from the cylindrical surface at Kn =
10"4 is shown in Fig. 8. 1 is the Knudsen layer, 2 is the Navier-Stokes region, 3 is the zone of supersonic expansion.
The temperature components on directions (Ty, Tr, Tz) and the average temperature T have qualitatively known
dependencies at the wall Tr < T< Tz= T9. The right (downstream the flow) Knudsen layer boundary is determined
by the value of the local Knudsen number KnL = (k/P)/(dP/dr) -0.01. Here P is a macroscopic parameter. It should
be noted here, as in the case described above by data of [13], that at Kn > 10"3 the transition through the sonic
velocity takes place under the single directed geometrical effect. It contradicts to the statement of classical gas
dynamics. But it is possible because of action of viscosity forces only at specific boundary conditions. In the result
of computations [22] it was shown that the use of the Navier-Stokes equations is reasonable only at Kn < 10"3 for the
region with M> 1 with boundary conditions determined by kinetic analysis.
Nonequilibrium processes in Knudsen layers by evaporation from the sphere [25] qualitatively are not different
from those described above for the cylinder. The quantitative difference is explainable. At the same Knudsen
number in the region 10"4 < Kn < I as the radial gradients of parameters are higher in the case of sphere, the relative
fraction of returning molecules is less and the distance from the evaporating surface to the point with M = 1 is also
less. In Fig. 9 the data characterizing the state of the gas on the surface with M = 1 in the range of the Knudsen
number from 10"4 to 10"2 are given. These data, having constant reduced macroscopic values in the wide range of the
Knudsen number, can be used as boundary conditions for continuum equations. The location of the surface with M =
1 is given in this figure as RI = 0.01 (r/r0 -1) / Kn.
4 0.6
?0~3 Kn
1,2 r/R
FIGURE 10. Isochores n/nw; nw is the saturation
FIGURE 9.1 is density; 2 is temperature; 3 is RI.
One more beneficial result of [25], which is worth to mention, is the proof of computational experiment of the
fact that the artificial prescription of the DFMV with arbitrary nonequilibrium for escaping of evaporated molecules
in the frames of conservation laws does not influence in essential on the sizes of the zone of relaxation (sizes of the
Knudsen layer).
It was shown in [26] that one can use the Knudsen criteria Knj, Kri2, Kn^ to describe the non-steady flow from a
spot of limited sizes, evaporated, for example, by laser irradiation. The determining linear sizes are, respectively, the
spot radius, the path of molecules with an average velocity in irradiation time, and the local linear scale on gradients
of macroscopic parameters. It is obviously that at Kn\ —> 0 the flow will be always plane-parallel one. At Kn2 —» 0
the one-dimensional flat parallel flow is realized in the vicinity of the surface. The Navier-Stokes region appears at
Kn2 < 0.002. The attainment of such Knudsen number Kn2 is possible at very high laser irradiation. For example, in
the case of a copper surface evaporated during 60 ns the irradiation intensity should be more than 106 W/cm2. At the
instant of start of the evaporation into vacuum the Knudsen layer has unlimited sizes. During the evaporation the
state of vapor near the surface tends to equilibrium in consequence of increasing a density and decreasing the local
gradient of parameters on the length of the mean free path. In the limiting case when the Kn2 —> 0 the usual viscous
transonic zone forms as it is described above for the cylindrical surface. Than the flow in the zone with M > 1 at a
low Knudsen number is governed by known laws of one-dimensional non-steady expansion of a gas into vacuum. At
a sudden stop of the irradiation the generated layer of a vapor is scattering in the form of a cloud, as it is shown in
Fig. 10. Here the picture of isochores is presented in the moment after 5 exposure times of laser irradiation for the
case Krii = 0.01, Kn2 = 0.177. The scattering of the cloud is going with the growth of nonequilibrium in the vicinity
of the surface. The formation of the Knudsen layer close to the surface, evaporating in the background with the
given pressure, is traced in [7], The evaluated from [7] conditions of appearance of the continuum flow region does
not contradict to the conclusions of [26] at the Knudsen number determined by prescribed initial data.
The problem of evaporation and condensation in ID flows of monatomic gases without physical and chemical
transformations, including the problem of inverted temperature gradient paradox, appears to date as worked out. The
excellent illustration of the completion of general studies of this problem is the new paper of Sone and his colleagues
[6], devoted to the transonic flow in the vicinity of the point M = 1, where the regions of evaporation and
condensation are brought to. The authors' address to very delicate field, concerning the fatal instability of phase
transitions, connected with reverse of a flow. At the present time the necessity of the generalization of all data
obtained in the traditional set of the problem has matured. It is important now for the practical engineering, taking
into account that more than twenty years have passed since researchers began to "arrange" [27] about the definition
of macroscopic boundary conditions.
The analysis of nonequilibrium processes in Knudsen layers of condensing and evaporating surfaces became
visible by use of the DSMC method. It allows using macroscopic boundary conditions with sufficient grounds. The
Knudsen layer close to the evaporating surface stretches for a few molecular mean free paths, number of which does
not depend practically on gas state parameters. But it is subjected to the influence of gradients of parameters, which
are conditioned by a flow geometry or current non-steady process of the flow formation. The nonequilibrium region
of the flow in the vicinity of surfaces with limited sizes can be distinguished in the flows with a determining
Knudsen number Kn < 10"3. The presence of the Knudsen layer is of specific boundary conditions, which allow the
transition through the velocity of sound due to the influence of viscous forces, when the geometrical effect has one
direction. The inverted temperature gradient close to the condensing surface is the result of the elevation of
temperature by collisions of opposite condensing and evaporating flows. It can be considered as a phenomenon in
continuum flow, resulted from the shock disturbance in the Knudsen layer at some peculiar conditions of the
At the present time studies of evaporation and condensation with essential physical and chemical transformations
and studies of processes when condensation coefficient depends on microscopic characteristics of molecular
collisions with a surface present a vast field of future activity.
1. Kucherov, R. Ya., Rickenglas, L. E., and Tsulaya, T. S., "Kinetic Theory of the Overcondensation for the Small Difference of
Temperature," J.Tech.Phys. 32, 1392-1397 (1962) (in Russian).
2. Labuntsov, D. A.,. "Analysis of Evaporation and Condensation Processes" Thermophys High Temperat. 5, 647 — 654 (1967)
(in Russian).
3. Anisimov, S. L, "On Evaporation of Metals Absorbing a Laser Radiation," /. Exper. Techn. Phys. 54, 339 — 342 (1968) (in
4. Sampson, R. E., and Springer, G. S., "Condensation on and Evaporation from Droplets by a Moment Method," /. Fluid
Mech. 36, 577-584 (1969).
5. Takata, S., Sone, S., Lhaillier, P., and Wakabayashi, M., "Evaporation from or Condensation onto a Sphere: Numerical
Analysis of the Boltzmann Equation for Hard-Sphere Molecules," Comp. Moth. Appl 35,193-214 (1998).
6. Sone, Y., Galse, F., Ohwada, T., and Doi, T., "Analytical Study of Transonic Flows of a Gas Condensing onto its Plane
Condensed Phase on the Basis of Kinetic Theory," Eur. J. Mech. B/Fluids 17, 277-306 (1998).
7. Sone, Y., and Sugimoto, H., "Strong Evaporation from a Plane Condensed Phase," in IUTAM Symposium, Gottingen, 1990,
pp. 293-304.
8. Kryukov, A. P., "Strong Subsonic and Supersonic Condensation on a Plane Surface," in Proceedings of the 17st Int.
Symposium on Rarefied Gas Dynamics, edited by A. E. Beylich, VCH, Weinheim; New York ; Basel; Cambridge, 1991, pp.
9. Kogan, M. N., and Abramov, A. A., "Direct Simulation Solution of the Strong Evaporation and Condensation Problem," ibid,
pp. 1251-1257.
10. Aoki, K., and Sone, Y., "Gas Flow around the Condensed Phase with Strong Evaporation or Condensation — Fluid Dynamic
Equation and its Boundary Condition on the Interface and their Application" in Advances in Kinetic Theory and Continuum
Mechanics, Paris, 1991, pp. 43-54.
11. Sugimoto, H., and Sone, Y., "Numerical Analysis of Steady Flows of a Gas Evaporating from its Cylindrical Phase on the
Basis of Kinetic Theory," Phys. Fluids A 4,419-440 (1992).
12. Sone, Y., and Sugimoto, H., "Kinetic Theory Analysis of Steady Evaporating Flows from a Spherical Condensed Phase into
Vacuum," Phys. Fluids A 5, 1491-1511 (1993).
13. Aoki, K., and Masukawa, N., "Gas Flows Caused by Evaporation and Condensation on Two Parallel Condensed Phases and
the Negative Temperature Gradient: Numerical Analysis by Using a Nonlinear Kinetic Equation," Phys. Fluids 6, 1379-1395
14. Pao, Y. P., "Application of Kinetic Theory to the Problem of Evaporation and Condensation," Phys. Fluids 14, 306 (1971).
15. Kogan, M. N., "Evaporation / Condensation Kinetics," in Proceedings of the 19^ Int. Symposium on Rarefied Gas Dynamics,
edited by J. Harvey et al., Oxford University Press, Oxford ; New York; Tokyo, 1995, v. 1, pp. 253-262.
16. Sone, Y., Ohwada, T., and Aoki, K., "Evaporation and Condensation of a Rarefied Gas between its Two Parallel Plane
Condensed Phases with Different Temperatures and Negative Temperature Gradient Phenomenon — Numerical Analysis of
the Boltzmann Equation for Hard Sphere Molecules," in Mathematics Aspects of Fluid and Plasma Dynamics, Lecture Notes
in Mathematics,^. 186-202 (1991).
17. Ytrehus, T., "The Inverted Temperature Gradient in Condensation — Revisited," in Proceedings of the 21st Int. Symposium
on Rarefied Gas Dynamics, edited by R. Brun et al., Cepadues-Editions, Toulouse, 1999, v. 1, pp. 471-478.
18. Plotnikov, M. Yu., and Rebrov, A. K., "Condensation on a Black Sphere. Knudsen Layer Structure," ibid, v. 1, pp. 487-494.
19. Sugimoto, H., and Sone, Y., "Numerical Analysis of Steady Flows of a Gas Evaporating from its Cylindrical Condensed
Phase on the Basis of Kinetic Theory," Phys. Fluids A 4,419-440 (1992).
20. Sone, Y., and Sugimoto, H., "Kinetic Theory Analysis of Steady Evaporating Flows from a Spherical Condensed Phase into
Vacuum," Phys. Fluids A 5, 1491-1511 (1993).
21. Bird, G. A., The Molecular Gas Dynamics, Clarendon Press, Oxford, 1976, pp. 25-30.
22. Rebrov, A. K., Plotnikov, M. Yu., and Bulgakova, N. M., "Formation of Supersonic Flow from a Cylindrical Source,"
Russian J. Eng. Thermophys.7, 81-97 (1997).
23. Knight, C. J., "Evaporation from Cylindrical Surface into Vacuum,"/. FluidMech.75, 469 (1976).
24. Sibold, D., and Urbassek, H. M., "Kinetic Study of Evaporation Flows from Cylindrical Jets," Phys. Fluids A 3, 870 (1991).
25. Bulgakova, N. M., Plotnikov M. Yu., and Rebrov A. K., "Modeling Steady-State Gas Expansion from a Spherical Surface
into Vacuum," Fluid Dynamics 32, 870-875 (1997).
26. Bulgakova, N. M., Plotnikov M. Yu., and Rebrov A. K., "The Study of Laser Evaporation Products Scattering by Method of
Direct Statistical Simulation," Thermophys. Aeromech. 5,421-429 (1998) (in Russian).
27. Soga, T., "On the Arbitrary Strong One-Dimensional Evaporation Problem," Trans. Jap. Soc. Aeron. Space Sci 21, 87-97