Non-steady Behavior of Vapor between Heater and Liquid
Surfaces
A. P. Kryukov, A. K. Yastrebov
Department of Low Temperatures, Moscow Power Engineering Institute,
Moscow, Russia
Abstract. Solution of the Boltzmann kinetic equation for non-steady heat transfer in vapor film between heater and liquid
was obtained. Knudsen number was varied from 0.005 to 0.1. Boundary conditions were different: given heat flux on the
heater surface or given temperature of this surface. It was shown and explained that condensation occurs during a nonsteady process, and steady mass flux is equal to zero. In the different applications of this problem variations of
macroscopic parameters (temperature of liquid surface and film thickness) are very small during non-steady stage.
Therefore vapor can be assumed to be steady in analysis of such applications.
INTRODUCTION
In the investigation of several two-phase systems there are some problems, in which strong deviation from
equilibrium state exists. Among them there is vapor film evolution at film boiling [1] and motion of highly thermal
conductive liquid in capillary with vapor in the presence of longitudinal heat flux [2].
Considering of non-equilibrium conditions is needed when film boiling of superfluid helium is studied. Heat
transfer in this liquid is very effective. Because thermal resistance of liquid is very small, non-equilibrium effects on
the inter-phase surface can be important. Kinetic treatment of vapor behavior in such a problem was used in [3]. The
calculation of recovery heat flux was subject of that paper. However it should be noted that only a steady problem
was studied in [3]. There are simple analytical expressions for steady heat and mass transfer through inter-phase
surface, but there are no such expressions for non-steady processes.
Because strong non-equilibrium takes place, the application of continuous matter mechanics methods is not
always correct and methods of molecular-kinetic theory are needed to solve these problems in part of vapor analysis.
In this paper non-steady heat and mass transfer processes in vapor film in application to such problems were studied
by numerical solving of the Boltzmann kinetic equation:
∂f r ∂f
+ξ r = J ( f )
∂t
∂x
(1)
Here f is the distribution function of vapor molecules, t is time, ξ is molecular velocity, x is coordinate, and J is
collision integral. The conservative discrete ordinates method [4] was used. Essential feature of this method is the
exact fulfillment of conservation laws. Collision integral for Maxwellian distribution is equal to zero exactly when
this method is used. The velocity distribution function is obtained as a result. The parameters of vapor (density,
pressure, temperature, mass and energy fluxes) are calculated as moments of this function.
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
PROBLEM
liquid
x
TI
L
vapor
TH
qH
0
heater
FIGURE 1. Scheme of problem.
The studied problem can be called “non-steady heat transfer in vapor film at film boiling”. The considered
system is shown in Fig.1. TH and TI are temperatures of heater and liquid surface respectively, qH is heat flux density
on the heater, x is coordinate, and L is thickness of the vapor film. Before t=0 all parts of the system (heater, vapor,
liquid) have the same temperature equal to TI, heat flux qH is zero. At the initial moment of time there is disturbance:
increase of heater temperature (TH=const, qH=var) or increase of heat flux on the heater (TH=var, qH=const). The
problem is considered as one-dimensional. Liquid surface temperature and film thickness were assumed to be
constants. The distribution functions for molecules moving from surfaces were Maxwellian functions:
f =
n
( 2πRT )
3
2
 ξ 2 + ξY2 + ξ 2Z 
exp  − X

2 RT


(2)
In this formula f is the distribution function, n is numerical density, T is surface temperature, R is gas constant,
and ξX, ξY and ξZ are velocity components. For surface of liquid temperature TI is given, and numerical density is
saturated vapor one. For heater surface at given TH numerical density can be found using condition of nonpenetrability (total mass flux density is equal to zero), at given qH values of n and T can be found using balance of
energy (energy flux in vapor is equal to qH) and mass.
The Boltzmann equation was solved as non-dimensional. Non-dimensional values were introduced as follows:
TN = TD TI
(3)
pN = pD pS (TI )
(4)
ρ N = ρ D ρ S (TI )
(5)
jN = jD ρ S (TI ) RTI
(6)
qN = qD pS (TI ) RTI
(7)
xN = xD < l > ( ρ I , TI )
(8)
t N = tD RTI < l > ( ρ I , TI )
(9)
Subscript “D” means “dimensional”, subscript “N” means “non-dimensional”. Subscript “S” means that value is
saturated vapor one. T is temperature, p is pressure, ρ is density, j is mass flux density, q is heat flux density, R is gas
constant, x is coordinate, <l> is mean free path, and t is time. Another non-dimensional parameters are not presented
because there are no data for them in this paper. Only non-dimensional values are used below, subscript “N” is not
written. Coordinate is introduced so that
Kn = 1 L
(10)
where Kn is Knudsen number calculated at initial state, and L is film thickness.
The macroscopic vapor parameters (density, pressure, temperature and mass flux density) are moments of the
distribution function and determined by the integration over three-dimensional space of molecular velocities (see for
example [5]). Non-dimensional parameters, which should be known before beginning to solve the Boltzmann
equation, are heater surface temperature or heat flux on the heater, film thickness, and the parameters of numerical
algorithm (time and coordinate steps, velocity space region in which the solution is being found, etc.).
RESULTS AND DISCUSSION
The quantitative results are presented in Fig. 2 – 9 as dependencies of vapor parameters (density, pressure and
temperature) on coordinate at different moments of time. Dependency of mass flux density at the liquid surface on
time is shown also. There are two different groups of presented results. The first group includes data for relatively
thick film (Kn=0.005), temperature of heater surface is given and equal to 2.3. The results of the second group are
for film of much lesser thickness (Kn=0.1), and these data were obtained at given heat flux on the heater (qH=0.3).
It is often assumed that evaporation occurs in such systems during non-steady and steady stages because the
surface of liquid is heated. But from another point of view, vapor temperature near the heater rises, thus vapor
pressure in this region rises also, and vapor begins to move towards the surface of liquid. Due to this reason vapor
pressure becomes greater than saturated vapor pressure at TI, thus vapor can condense during the non-steady process.
Mass flux does not depend on coordinate in steady state. Heater surface is not penetrable, mass flux on it is always
zero. Therefore mass flux in steady state is equal to zero. Authors’ opinion is that evaporation can occur when vapor
film is connected with vapor over free surface of liquid, or when liquid surface is not stable. In the second case
vapor bubbles can form and remove vapor from inter-phase surface. During the non-steady process liquid can
evaporate also when its thermal conductivity is low and surface temperature rises very quickly. However,
evaporation can not happen at steady stage when statement of problem is such that studied here.
1,1
1,0
Density
0,9
0,8
0,7
t=110
t=415
steady state
0,6
0,5
0,4
0
50
100
150
C oordinate
200
100
C oordinate
200
FIGURE 2. Density for TH=2.3 and Kn=0.005.
1,20
t=110
t=415
steady
state
Pressure
1,16
1,12
1,08
1,04
1,00
0
FIGURE 3. Pressure for TH=2.3 and Kn=0.005.
50
150
Temperature
2,2
2,0
t=110
t=415
steady state
1,8
1,6
1,4
1,2
1,0
0
50
100
150
Coordinate
200
FIGURE 4. Temperature for TH=2.3 and Kn=0.005.
Mass flux density
0,12
0,10
0,08
0,06
0,04
0,02
0,00
0
500
1000
Tim e
1500
2000
FIGURE 5. Mass flux density on the liquid surface for TH=2.3 and Kn=0.005.
1,1
1,0
Density
0,9
0,8
0,7
0,6
0,5
0
FIGURE 6. Density for qH=0.3 and Kn=0.1.
2
4
6
C oordinate
t= 2
t= 10
t= 30
steady
state
8
10
1,14
1,12
Pressure
1,10
1,08
1,06
t=2
t=10
t=30
steady state
1,04
1,02
1,00
0,98
0
2
4
6
C oordinate
8
10
FIGURE 7. Pressure for qH=0.3 and Kn=0.1.
2 ,2
t= 2
t= 1 0
t= 3 0
s te a d y s ta te
Temperature
2 ,0
1 ,8
1 ,6
1 ,4
1 ,2
1 ,0
0
2
4
6
C o o rd in a te
8
10
80
100
FIGURE 8. Temperature for qH=0.3 and Kn=0.1.
0,12
Mass flux density
0,10
0,08
0,06
0,04
0,02
0,00
0
20
40
Tim e
60
FIGURE 9. Mass flux density on the liquid surface for qH=0.3 and Kn=0.1.
In both groups of results pressure in steady state is near to independent on coordinate. When heat flux is given
pressure increases almost always during non-steady stage. At the given heater temperature pressure firstly rises, then
lowers. Mean vapor density is constant when there is no variation of vapor state near liquid surface, then this value
begins to lower. Temperature of vapor always rises in both cases. During non-steady process mass flux density on
liquid surface is zero or greater than zero, thus vapor condensation occurs. This value in steady state is not equal to
zero exactly, it occurs because the numerical solution error takes place. Hence the presented above qualitative
description of heat and mass transfer processes is in agreement with quantitative results.
Maximal value of mass flux density on the liquid surface depends on Knudsen number weakly. For TH=2.3 and
Kn=0.1 (these results are not presented here in figures but they were obtained) this value is about 0.17, in shown data
(Fig. 5) it is about 0.11. However values of non-steady process duration, during which non-zero mass flux exists, can
differ substantially. For Kn=0.1 this duration is about 100, for Kn=0.005 it is about 2000.
Solution for some value of heat flux includes dependency of heater temperature on time. THS is heater
temperature in steady state (for example, at qH=0.3 and Kn=0.1 THS=2.3). Steady state results of this problem solving
for given heater temperature equal to THS do not differ from results for respective value of qH (difference less than
1% is realized as result of numerical solution error). Vapor heating at given heater temperature is more intensive
than at given heat flux on the heater. Hence at a given heater temperature vapor temperature in the beginning of
process increases more quickly.
The duration of non-steady process is very small in a macroscopic scale. For example, if a considered substance
is helium at TI=2 K, then this time for non dimensional TH=2.3 and Kn=0.005 would be about 1 microsecond, for
qH=0.3 and Kn=0.1 it would be about 0.05 microsecond. Mean free path of helium atoms at TI=2 K is about 0.03
micron, thus in this case non-dimensional film thickness L=200 corresponds to dimensional film thickness about 6
microns. In analogy for L=10 dimensional film thickness is about 0.3 micron. The investigation of applications [1, 2]
shows that liquid surface velocity has such value that displacement of liquid surface during this time is much less
than film thickness. Liquid helium at this temperature is superfluid therefore heat transfer in liquid is very effective
and variation of liquid surface temperature during the non-steady process cannot be large and cannot affect vapor
behavior. Due to these reasons assumptions about constant values of film thickness and liquid surface temperature
are correct. Note that boundary surfaces can be curved in application problems: in [1] both surfaces are cylindrical,
in [2] heater surface is flat and liquid surface is near to spherical. Application of obtained for flat surfaces results to
such problems is correct if film thickness is much less than radius of curvature.
Steady state results were compared with equations for heat transfer through inter-phase surface (they are
presented, for example, in [1,2,6]):
p − pS (TI ) = 0.44q
2 RTI
 p
 RTI
q = 8 p 
− 1
 pS (TI )  2π
(11)
(12)
These equations were obtained using moment method of the kinetic Boltzmann equation solving. Equation (11)
is linear and it is correct if heat flux is small, equation (12) is nonlinear and more general. Here q is heat flux on the
surface, p is vapor pressure near the surface, TI is surface temperature, pS(TI) is saturated vapor pressure at TI, and R
is gas constant. Both these equations are dimensional. Non-dimensional heat flux as it was introduced in formula (7)
is
q = 3.21( p − 1)
(13)
q = 8 p ( p − 1)
2π
(14)
In results for qH=0.3 and Kn=0.1 vapor pressure near the liquid surface is equal to 1.071. Heat flux calculated
using linearized equation (13) is 0.228, if nonlinear equation (14) is used this value is 0.243. Heat flux on the surface
in obtained numerical solution is 0.245; difference between moment method results and presented results is about
7% in the first case and about 0.8% in the second case. The same calculations were carried out for TI=2.3 and
Kn=0.005. Pressure near the surface of liquid is 1.018 in this case, heat flux values found from (13) and (14) are
identical and equal to 0.058, heat flux found numerically is 0.055, difference is about 5%. Thus presented results are
in good enough agreement with the previous study of steady heat transfer through inter-phase surface in nonequilibrium conditions.
CONCLUSION
It was shown qualitatively as well as quantitatively that vapor condensation occurs during the non-steady
process. Steady mass flux obtained as a result of numerical solution is not equal to zero exactly due to calculation
error. Maximum of mass flux density on the liquid surface depends on Knudsen number weakly: 0.17 at Kn=0.1 and
0.11 at Kn=0.005 (boundary conditions are identical). The duration of the non-steady process is very small in a
macroscopic scale, therefore variations of film thickness and liquid surface temperature are negligible. However
these variations can be not neglected if Knudsen number is very small or liquid surface velocity is large.
ACKNOWLEDGMENTS
The work is supported by Russian Foundation for Basic Research (projects 02-02-16311, 02-02-06152 and in
part 00-15-96543). The authors are grateful to Professor F. G. Tcheremissine (Computing Center of Russian
Academy of Sciences) for consultations concerning the Boltzmann equation solving method.
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