Using a Thin Wire in a Free-Molecular Flow for
Determination of Accommodation Coefficients of
Translational and Internal Energy
A. K. Rebrov, A. A. Morozov, M. Yu. Plotnikov, N. I. Timoshenko, A. V.
Shishkin
Institute of Thermophysics, Novosibirsk, Russia, E-mail:[email protected]
Abstract. A new experimental-computational method for determination of the translational and internal energy
accommodation coefficients on a thin wire in a free-molecular flow is elaborated. The base of the method is a numerical
solution of the heat balance equation of the wire. Measurements were performed in a low-density wind tunnel with Ar,
He, H2, N2, CH4, CO2. Besides, the proposed method provides determination of thermophysical properties of the wire.
I. INTRODUCTION
The accommodation coefficient (AC) of energy is determined as ratio
α=
Ei − E r
.
Ei − E w
(1)
Here Ei and Er are the energies of incident and reflected molecules and Ew is the average energy of reflected
molecules corresponding to the surface temperature. It is this value usually used as a measure of energy exchange
between gas and surface from the times of M. Knudsen [1]. Such a definition assumes an equal accommodation for
the translational and internal energy. By using in formula (1) the translational (internal) energy, one can obtain the
definition for partial accommodation coefficients of translational (internal) energy α′ (α″). Some theoretical and
experimental results show that AC's of translational and internal energy may differ considerably [2 – 4].
Probing with a thin wire has been widely used at high Knudsen numbers for measurements of total energy AC’s
in the quiescent gas [5]. The attempt to measure partial AC’s for translational and internal energy of polyatomic
molecules in the quiescent gas on early fifties was called in questions later in [6]. The theory of energy AC
calculations in studying heat transfer of a cylinder in a free molecular flow was essentially elaborated in [7]. Using
of a thin wire anemometer for diagnostics of rarefied gas flows in combination with other instruments allowed to
determine the total AC on the base of simplified calculation of wire heat balance [8]. It was shown in [9, 10] that
idea of thorough analysis of a heat balance for cylinder in a free molecular flow can work at the determination of
partial AC's for translational and internal energy. In these works the AC's were determined using an analytical
solution of a simplified heat balance equation proposed in [11]. Those simplifications are unacceptable for a wide
range of the wire temperature. They bring essential uncertainties into analysis. Our paper presents results of
determination of AC's by accurate numerical solving the heat balance equation with consideration of influence of
electro-thermophysical parameters of the wire.
II. METHOD
The heat balance for the wire of a unit length is defined by the following equation:
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
1016
cgπr02
∂Tw I 2 ρ
d 2Tw
+ 2πr0 h(Tr − Tw ) − 2πr0 εσ(Tw4 − T∞4 ) .
= 2 + λπr02
2
∂t
πr0
dx
(2)
Here x is the distance measured along the wire; Tw is the wire temperature; r0 is the radius; c is the specific heat
capacity; g is the density; I is the electric current; ρ is the specific electric resistance; λ is the heat conductivity
coefficient; ε is the surface emissivity; σ is the Stefan-Boltzmann constant; T∞ is the temperature of surrounding
surfaces; Tr is the recovery temperature of the wire; h is the heat transfer coefficient for the given flow parameters.
The dependencies of ρ, ε and λ on the temperature are taken as ρ = ρ0 (1 + α0T), λ = λ0 (1 + β0T), ε = ε0 (1 + χ0T).
The values ρ0, ε0 and λ0 correspond to T = 0˚С.
The boundary condition for the equation (2) is the wire-holder junction temperature Ts. A correct determination
of this temperature is one of the main difficulties in solving the considered problem. It is common for the holder to
be massive comparing to the wire. If the holder temperature Tb at a large distance from the junction is measured or
evaluated, one can consider Ts depending on the wire-to-holder heat flux and the holder heat resistance from the
junction to the surface with the known temperature Tb. The heat transfer at small distances from the junction (r ≤ r0)
is assumed to be as in a plane layer and at large distances (r ≥ r0) as in a spherical layer. It allows us to obtain a
simple approximate evaluation of Ts:

q
Ts ≈ Tb +
2πλ b

 1 1 
q
 −  +
.
 r0 rb  πλ b r0
Here q is the heat flux from the wire to the holder (that is determined in calculation through the temperature
gradient at the wire end); λb is the heat conductivity coefficient of the holder; rb is the holder radius. By rb >> r0 one
can obtain
Ts ≈ Tb +
3 q
.
2 πλ b r0
To solve the equation (2) it is necessary to know the heat transfer coefficient h and the recovery temperature Tr.
The kinetic theory of gases allows to find these quantities at convex surfaces in a free-molecular flow [12]. The
expressions for h and Tr for a cylinder located perpendicular to the flow are given in [9] for a gas with internal
degrees of freedom:
h=
1
( ) US ⋅ [I
nkξ ⋅ exp − S2 ⋅
2
0
]
+ S 2 (I 0 + I 1 ) ,
(3)
4 π

1
T 
) + α ′′j  .
Tr = ⋅ α ′(2 S 2 + 5 −
2
ξ 
1 + S (1 + I 1 / I 0 )

(4)
Here n, U, T are the number density, velocity, and static temperature of the gas flow; S = U/(2kT/m)1/2 is the
speed ratio; m is the molecular mass; k is the Boltzmann constant; ξ = 4α′+α″j; j =
5 −3 γ
γ −1
is the number of internal
degrees of freedom; γ is the adiabatic exponent; I0 and I1 are the modified Bessel functions of zero and first order for
argument S2/2.
It is worth noting that in a quiescent gas (U = 0) the equation (4) is reduced to the form of Tr = T, and the wire
resistance depends only on complex ξ = 4α′ + α″j of the equation (3). So it is possible to separate the AC's of
translational and internal energy only in the case when there is a directional movement of gas.
In all cases of determination of AC's, we used the trial-and-error method with thoroughly minimization of errors.
A series of experimental measurements of the wire resistance R′k is performed by different currents Ik, k = 1..N.
Scanning of all suitable pairs of AC's (α′, α′′) has been accomplished. For each such a pair (α′, α′′) and for each
current Ik the equation (2) is numerically solved and the temperature distribution along the wire Tw(x) = Tw(x, α′,
α′′, Ik) is found. Given the temperature distribution, the integral wire resistance is calculated R(α′, α′′, Ik ) =
ρ0 L
(1 + α 0Tw ( x, α′, α ′′, I ) )dx (L is the wire length). Then the discrepancy between the computed and experimental
πr02 ∫0
resistance ∆ k = R(α ′, α ′′, I k ) − Rk′ is calculated, and the function F(α′, α′′) =
∑
N
1
∆2k is considered. As the sought-
for AC's, such a pair of (α′, α′′) is chosen for which F(α′, α′′) takes the minimum value. Such an approach allows
reducing the influence of uncertainties, which are inevitable in experiments.
1017
III. EXPERIMENTS, RESULTS AND DISCUSSION
The experiments were performed in a low-density wind tunnel, where rarefied gas flow was created by free jet
expansion of a gas through a sonic nozzle. The nozzle diameter is d∗ = 6 mm. The stagnation pressure is P0 = 1000 ÷
2100 Pa, the background pressure is about 2.7 Pa. The stagnation temperature is T0 = 290 – 300 K. A thin wire probe
was placed on the jet axis at distances of 2 ÷ 7 d∗ from the nozzle exit. Given such experimental conditions, the flow
near the wire is free from influence of the background gas [13]. So the flow parameters on the jet axis for the
considered distances correspond to those calculated for expansion of nonviscous gas [14]. It was confirmed by
results of numerical modeling of viscous gas expansion that was graciously presented by P. A. Skovorodko.
A gilded tungsten wire with the diameter of 8.3 µm was used as the probe. With such a diameter, a free
molecular flow was provided at the above mentioned conditions. The wire was welded to a stainless steel holder in
the form of a cylinder with the diameter 0.6 mm tapered to the size 0.2 mm. The typical wire length L was of 3 – 10
mm.
One of the most important requirements to experiments in the wind tunnel and calibration procedures is a high
precision of measurement of the voltage and amperage. These strict requirements are inevitable, as well as main
results are extracted by precise numerical calculation of the temperature distribution along the wire. A voltmeter and
a source of electric current with accuracy of 0.1 % were used for our measurements.
The experiments in a free jet were preceded by determination of electro-thermophysical properties of the wire:
the specific resistance ρ, the heat conductivity coefficient λ, and the emissivity ε, with taking into consideration their
dependence on temperature. In these experiments, the wire tips were clamped between solid copper terminals to
provide an ideal heat contact between the wire and the terminals as much as possible. The dependence ρ = ρ(Tw) is
determined by the measurements in a thermostat in air by atmospheric pressure. When measuring electric resistance
in the thermostat, a low amperage is used to eliminate the electric heating of the wire.
The surface emissivity ε(Tw) is found by preliminary calculation (by solving the equation (2) for vacuum) of
dependence R = R(ε) for the wire with L = 100 mm for two constant values of the heat conductivity coefficient λ of
120 and 150 W/(m·K). For such a length, the heat flux through radiation is by one order higher than through the
holders. The range of λ reasonably covers the spread of these values known from the literature. The emissivity ε is
determined by the value of R known from the experiments. The emissivity determined by this means is used to
calculate a more definite heat conductivity coefficient with the length wire L = 10 mm, when the heat flux from the
wire to the holder dominates. After this, the emissivity can be defined more accurately from the determined heat
conductivity.
In the results of these calibration procedures, the following characteristics for the tested wire were determined:
ε = 0.05 ⋅ [1 + 0.00335 ⋅ (Tw − 273)] ,
ρ = 5.65 ⋅ 10 −8 [1 + 0.0041 ⋅ (Tw − 273)] Om ⋅ m ,
145 − 0.11 ⋅ (Tw − 273) W/(m ⋅ K), Tw ≤ 250°C
λ=
117 − 0.06 ⋅ (Tw − 523) W/(m ⋅ K), Tw ≥ 250°C.
At the first stage, the AC's were determined for monoatomic gases Ar and He. One of the typical experimental
dependencies of the wire resistance on current is presented in fig. 1 (solid line) for argon flow (L = 3.45 mm, S =
5.84). By looking through the appropriate AC's, a minimal discrepancy between experimental and computational
dependencies for this case has been obtained for α′ = 0.85. The final fitting of computational dependence (dashed
line) to the experimental one is shown in fig. 1.
In experiments with polyatomic gases (H2, N2, CH4, CO2) performed at the room stagnation temperature and
temperature of the wire below 400°C, only rotational degrees of freedom as internal ones are active. So the above
procedure was performed for a set of pairs of AC's (α′, α′′). The figure, presenting discrepancy ∆ =
1
∑
N
∆2 in
N 1 k
dependence on AC's, is three-dimensional one. Fig. 2 shows projection of this figure on (α′, α′′) plane for nitrogen
flow (L = 9.4 mm, S = 4.64).
1018
R,
Om
experiment
computation
6
5
4
3
0
I, mA
20
10
FIGURE 1. The final fitting of computational dependence of the wire resistance R on the current I to the experimental one for Ar
by α′ = 0.85 (P0 ≈ 2100 Pa, L = 3.45 mm, S = 5.84).
α′′
0.60
0.4
2
0.3
0.3
5
0.58
0.2
5
0.212
0.2
06
0.2
03
0.56
0.3
0.225
0.54
0.4
2
0.52
0.62
0.64
0.3
5
0.66
0.68
0.70
α′
FIGURE 2. The field of discrepancy ∆ of experimental and computational dependencies of the wire resistance R by
determination of AC's for N2 (P0 ≈ 2100 Pa, L = 9.4 mm, S = 4.64; α′ = 0.67, α′′ = 0.57).
1019
AC’s averaged over many measurements with different wire probes, gas pressures, and at different distances
from the nozzle are presented in the table I. The comparison of results of different research groups for AC’s is
usually questionable, since it is difficult to find the results for identical or similar conditions on many key parameters
(the energy of incident molecules, interaction angle, original structure of the surface, characteristics of the surface
sorbate, the surface temperature). This task is completely indefinite for surfaces of engineering interest. In table II,
some data from literature is given at least at close interaction energy. Comparison of tables I and II shows qualitative
agreement for our data and those from literature.
TABLE I. Accommodation coefficients.
Gas
He
Ar
N2
CH4
CO2
H2
α′
0.29 ± 0.05
0.8 ± 0.05
0.67 ± 0.04
0.78 ± 0.1
0.9 ± 0.23
0.44 ± 0.04
α′′
—
—
0.56 ± 0.08
0.73 ± 0.16
0.79 ± 0.15
-0.15 ± 0.05
TABLE II. Data on AC's from literature.
Gas
He
He
Ar
Ar
N2
N2
CH4
CO2
Sufrace
W, covered by O
Pt
W, covered by gas
Pt, burning in vacuum
W, burning in vacuum
Au, covered by gas
Pt
W, covered by CO2
Tw
330
310
300
410
450
293
373
330
α′
0.18
0.24
1.0
0.76
0.36*
0.89
0.81*
0.99*
α′′
—
—
—
—
—
0.78
—
—
Reference
[15]
[16]
[17]
[16]
[18]
[4]
[17]
[15]
* AC obtained for total energy.
From the presented above experiments for N2, CO2 and CH4, the conclusion can be made that for low energies of
molecules (corresponding to the room stagnation temperature) the AC’s of translational energy are slightly higher
than the AC’s of internal energy. It is confirmed by known theoretical [3] and experimental [4] works.
Determination of AC of internal energy is of particular interest in the proposed method. The available data on
such AC’s are very scanty. It is known that the molecular beam facilities are main instruments for experimental
study of the gas-surface interactions on the level of information on translational and internal energy distribution
function. In numerous molecular beam experiments, AC is not presented usually, and often there is not enough
experimental data for calculation of those coefficients. Besides, the most interesting experimental data for molecular
beams are obtained for conditions with well-cleaned oriented crystal surface.
Analysis of the papers where the authors provided enough data for calculation of the AC revealed that those
coefficients for translational and internal energy might be different significantly. So at a very high translational
energy of incident molecules, the rotational AC can be much higher than unity [19, 20]. The effect may be
accounted for translational-rotational energy transition during inelastic molecule-surface interaction.
This challenge to the traditional definition of internal energy AC is forced by our experiments with hydrogen.
The peculiarity of the hydrogen jet adiabatic expansion at our conditions (d∗ = 6 mm, P0 = 1300 ÷ 1400 Pa, T0 = 290
– 300 K) is the freezing of the rotational energy just close to the nozzle exit. The probe was placed at distance of 4d∗
from the nozzle exit. The obtained negative value of rotational AC has been verified by reiterated experiments with
different computational treatment. The qualitative stability of results has led to persuasion that the transition of
frozen rotational energy at the lowest energy levels to the translational energy of reflected molecules is responsible
for the intriguing result. The indirect evidences of the explanation were found in [21, 22]. Studying of molecular
hydrogen interaction with the surface (001) MgO, the authors of [21] have observed scattering corresponding to
deactivation of rotational energy on definite low levels by inelastic gas surface collisions. In [22], partial conversion
of rotational energy into kinetic energy of the scattered molecules by interaction of NO molecules with graphite
surface was shown.
1020
On the base of all the said, the general conclusion on internal energy transitions at the collision with a surface can
be expressed. The definition of this process, attendant by rotational-translational energy coupling, as accommodation
process contradicts to physical content. At least for polyatomic gases, the definition "transformation coefficient"
(TC) of energy, which is not limited by the bounds 0 – 1, quite fits for scientific literature.
III. CONCLUSION
The vast variety of experimental data on accommodation coefficients is very insufficient to cover the practical
needs in operation with an engineering surface. That is why the elaboration of this method for study of translational
and internal energy exchange by gas–surface interaction under conditions similar to reality of technologies is of
paramount interest. The proposed method is quite simple and effective tool in receiving of data useful both for
comprehension of gas – surface energy exchange and for simulation of gas – surface interaction in real technological
applications.
ACKNOWLEDGMENTS
This work was supported by INTAS (grant № 99–0749).
REFERENCES
1. Knudsen, M., Ann. Physik, 34, 593 (1911).
2. Legge, H., “Recovery Temperature Determination in Free Molecular Flow of a Polyatomic Gas,” In Rarefied Gas Dynamics.
dited by H. Oguchi, Proc. of the 14th Intern. Symp., v.I, 1984, pp. 271-278.
3. Feuer, P., J. Chem. Phys., 39, 1311 (1963).
4. Schafer,K., and Riggert, K. H., Zs. Elektrochem. Ber. Bunsenges. Phys. Chem. 57, 751 (1953).
5. Wachman, H. Y., “Thermal Accommodation Coefficient: The Contribution of Lloyd Brewster Thomas,” in Rarefied Gas
Dynamics: Experimental Techniques and Physical Systems. Progress in Astronautics and Aeronautics, ed. by B. D. Shizgal and
D. P. Weaver, v.158, 1994, pp. 479-493.
6. Petek, B., and Kuscer, I., ”Can Partial Accommodation Coefficients be Measured?” in Rarefied Gas Dynamics, ed. by R.
Campargue, Proc. of the 11th Intern. Symp., v.II, 1979, pp.1399-1403.
7. Tsien, H. S., J. Aeronaut. Sci.,15, N 10 (1948).
8. Gottesdiner, L., J. Phys. E: Sci. Instrum., 13, 908-913 (1980).
9. Bulgakov, A. V., and Prikhodko, V. G., J. Appl. Mech. Tech. Phys., 5, 126 (1986) (in Russian).
10. Bulgakov, A. V., Prikhodko, V. G.,Rebrov, A. K. and Skovorodko, P. A., “Free Jet Expansion of Heavy Hydrocarbon Vapour,”
in Rarefied Gas Dynamics: Physical Phenomena. Progress in Astronautics and Aeronautics, Proc. of the 16th Intern. Symp., v.
117, 1988, pp. 92-103.
11. Lord, R.G., J. Phys. E.: Sci. Instrum., 7, 56 (1974).
12. Kogan, M.N., Rarefied gas dynamics, Plenum Publ. Co., New York, 1969.
13. Rebrov, A.K., J. Vac. Sci. Technol. A, 19, 1679 (2001).
14. Zhohov, V.A., and Homutsky, A.A., Atlas of supersonic flows of freely expanding ideal gas flowing out of axisymmetric
nozzle, TsAGI, Moscow, 1970 (in Russian).
15. Wachman, H. Y., Thesis, University of Missouri, Columbia, 1957.
16. Reiter F. W., Camposilvan J., and Nehren R., Warme- und Stoffubertrag., 5, N 2 (1972).
17. Von Ubisch H., Appl. Sci. Res., A 2, 364 (1949–1950).
18. Chen S.H.P., Saxena S.C., Int. J. Heat Mass Transf., 17, 185 (1974).
19. Lahaye, R., J., Stolte W.E., Holloway, S., Klein, A.W., J.Chem. Phys., 104, 8301-8311 (1996).
20. Novopashin, S., Dankert, C., Lehmkoster K., Mohamed A., and Pot T.,"Rotational Temperature Measuements of N2 Molecules Reflected from a Wall," in Rarefied Gas Dynamics, edited by Ching Shen, Proc. of the 20th Intern. Symp., Peking
University Press, Beijing, China, 1997, pp. 428-433.
21. Rathbun, L. and Ehrlich, G., “Extremal Scattering of Molecular Hydrogen from (001) MgO,” in Rarefied Gas Dynamics, edited
by J. L. Potter, Proc. of the 16th Intern. Symp., 1977, pp.555-576.
22. Hager J., Glatzer D., Kuze H., Fink M., and Walther H., Surf. Sci., 374, 181 (1997).
1021