22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Transport properties of helium plasma at atmospheric pressure
A. Mahfouf1, P. André1, G. Faure1 and M.-F. Elchinger2
1
Blaise Pascal University, LAEPT, Clermont Ferrand, 24 avenue des Landais, FR-63171 Aubière, France
2
Limoges University, SPCTS, CNRS, 123 avenue Albert Thomas, FR-87060 Limoges, France
Abstract: Transport properties of thermal helium plasma used generally in blown plasma
jet for spray process were calculated according to the Chapman-Enskog theory. The
electrical conductivity, the viscosity, the thermal conductivity and combined diffusion
coefficients are computed from room temperature up to 50000 K at different pressures
assuming local thermodynamic equilibrium. The plasma composition is calculated using
the Gibbs energy minimization method. The helium neutrals, electrons, singly and doubly
ionized helium species have been taken into account in our calculations.
Keywords: transport coefficients, helium plasma, viscosity, thermal conductivity,
electrical conductivity
1. Introduction
The knowledge of transport coefficients is an essential
prerequisite for numerical simulation of thermal plasmas.
Many industrials applications that use the helium plasma
as spraying process [1] need accurate computation at high
temperature. The estimation of transport coefficients
(viscosity, thermal conductivity, electrical conductivity
and diffusion coefficient) is very difficult to do
experimentally at high temperature, thus the
theoretical/numerical calculation is the preferred method.
Two approaches solutions of the Boltzmann
integro-differential equation have been proposed by Grad
[2] and Chapman-Enskog [3] using the analysis of
rigorous kinetic theory. The first is called Grad's moment
method, the second Chapman-Enskog method. This last
is the most commonly used to calculate transport
coefficients. We opted for it in this paper.
2. Collision Integrals
The transport of matter, momentum and energy is
essentially due to the collision between particles and the
existing force fields.
The study of interparticular
interaction is crucial to understand the phenomena of
transport. In the microscopic world, the notion of
collision between atoms or molecules is interpreted as an
interaction between the forces associated with each of the
interacting particles. Throughout this paper we suppose
that the plasma is in low-density, thus we can neglect the
three-body interaction and take into account only elastic
collisions.
2.1. Binary collision
We consider an elastic collision between two particles
having masses m and M, separated by a distance r. b, r min
represent the impact parameter and the distance of closest
approach between the colliding particles, respectively.
The angle of deflection χ is defined by the following
equation:
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∞
χ (b, E ) = π − 2b ∫r min
dr
r2
v(r ) b 2
− 2
1−
E
r
(1)
E is the total energy of the system of particles m and M.
v(r) is the interaction potential between particles and is
the most interesting parameter for the study of the
collision problem [2].
2.2. Interaction potential energy
There are many forms of interaction potentials energy
to describe the same interaction, therefore, it is hard to
choose the good potential for a given interaction.
However one can select it according to the nature of the
interaction force (repulsive or attractive). We can show in
following examples some types of interactions that can be
found in thermal plasma:
a. Charged ↔ Charged: This interaction is governed by
the Coulomb interaction potential [4], but according to the
assumption of quasi-neutrality of the plasma [5], a
shielding effect must be taken into account.
Consequently this binary interaction will be described by
a Coulomb potential shielded at the Debye length λD [5]:
V ( r ) = ±V0
with: V0 =
qe
λD
r
−r

exp
 λD 
(2)
2
4πe 0 λ D
A sign ( ± ) depends on the sign of the particle charge
(electron-ion). The different formulas of Debye length
are discussed in is discussed in [12].
b. Neutral ↔ Neutral: There are a lot of interaction
potentials susceptible to describe this interaction, such as
Lennard-Jones potential, Morse potential, Buckingham
1
Potential and others. We have opted for the HulbertHirschfelder potential energy curve to describe this
collision because it depends only on the spectroscopic
constants [4].
c. Neutral ↔ Charged: The collision between the neutral
and the ion can be studied using the same interaction as
neutral-neutral.
d. Neutral ↔ Electron: In most cases, it is necessary to fit
experimental data in order to find an adequate potential
for this interaction.
2.3. Collision cross-section
In the case of the elastic collision, one can define the
total collision cross-section Q(E) as:
∞
Q ( E ) = 2π ∫ (1 − χoσ( χ ) ).bdb
0
π
(3)
χ
γ
∞
∫e
−( γ ) 2
(γ ) 2 s +3 Q ( E ) dγ
(7)
0
= E/kT and Q(E) is seen in the previous section.
 , are the orders of approximation, k is the Boltzmann
constant and µ the reduced mass of the colliding particles.
3.2. Viscosity
The viscosity η ( kg.m s ) is the transport coefficient
related to gradient in velocity expressed as follows:
1
1
(4)
is defined in Eq.(1).
<Ω
( 2, 2 )
(T ) >
(8)


J = −α∇ϕ
(5)

where J the flux vector is appropriate to quantity ϕ , and
α is a proportionality constant known as the transport
coefficient. The negative sign applies because the
quantity is transported from regions of high concentration
to regions of low concentration.
3.1. Coefficient of self-diffusion
In the case of gradient in concentration, the transport
coefficient is called the diffusion coefficient D (m.2s-1).
Chapman-Enskog have formulated all transport
coefficients, Eq.(6), Eq.(8) and Eq.(9), in term of collision
integrals [2], [3], Eq.(7), the coefficient of self-diffusion
of pure gas was formulated in a first approximation as
follows:
T3
D (T ) = 2.628 × 10 −7 
M
The flux correspondent is the viscous stress tensor [1].
3.3 Thermal conductivity
In the case of gradient in temperature, the quantity ϕ
transported is the energy which is related to heat flux
vector Eq.(5) through the thermal conductivity
κ ( J .m −1 .s −1 .K −1 )
3. Transport Coefficients of Pure Gases
Gradient in concentration, velocity and temperature,
causes a net transport or a net displacement of mass,
momentum, and energy, respectively [5]. We can
formulate mathematically this notion of transport as
general form of a flux:
1
2

1
(6)

(1,1)
(T ) >
 < pΩ
where M, p and T are the molecular weight of the particle,
the pressure in atmosphere and temperature in Kelvin,
2
with
kT
2πµ
is the differential scattering cross-section:
b
db
σ (χ , E) =
σin( χ ) dχ
and
Ω  ,s (T ) =
2
η (T ) = 2.6693 × 10 −6 (MT )2
0
σ (χ , E)
(1,1)
−1 −1
= 2π ∫ (1 − χoσ( χ ) ).σ ( χ , E ) σin( χ ) dχ
with
o
respectively. < Ω ( A ) > is the reduced diffusion
collision integral and is was formulated by Hirschfelder
[2] as:
expressed as:
1
1
 T 2
(9)
κ (T ) = 8.3227 × 10 

( 2, 2 )
(T ) >
M  <Ω
−5
We remark that all transport coefficients Eq.(6), Eq.(8)
and Eq.(9) are expressed in function of so called collision
integrals see the Eq.(7), and the integral collision is
function of collision cross section see the Eq.(3) which is
itself function of angle of deflection see the Eq.(1). In
summary, to compute the coefficients returns to evaluate
efficiently three integrals which are included one into the
other one. We have developed a numerical program
which compute the three integrals in same time using the
Clenshaw-Curtis quadrature [6] and we have successfully
eliminated the singularities [7] that occur in collision
cross section. Our program has been used to calculate the
transport coefficients of the helium gas for temperature
below 10000 K, the results in Table 1 are compared with
those of literature [8] and they are in good agreement.
Only helium neutrals have been taken into account up to
this temperature (10000 K).
4. Plasma Equilibrium Compositions
At high temperature (T > 10000 K) we have to take
account all interactions which may occur between
neutrals (He), ions (He+, He++) and electrons (e-). To
determine the proportion of each species in the helium
plasma at each temperature we have computed the
equilibrium compositions of dissociated and ionized
helium for different pressures using the Gibbs energy
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Table 1. Variation of transport coefficients of helium gas
with temperature.
T
(K)
300
500
1000
2000
4000
5000
7000
10000
η
κ
(10-3.W.m-1.K-1)
154.904
221.117
363.106
606.99
1036.420
1237.182
1626.459
2188.097
(10-6.Pa.s)
19.885
28.276
46.613
77.922
133.049
158.822
208.795
280.895
show the temperature dependence of the electrical
conductivity of helium plasma in Fig. 2.
D
(10-4.m2.s-1)
1.705
4.101
13.661
46.349
160.755
241.290
447.861
865.839
minimisation method [9]. Fig. 1 presents the variations of
the helium plasma compositions as a function of
temperature for different pressures.
Fig. 2. Evolution of electrical conductivity of partially
helium plasma with temperature for different pressures.
The electrical conductivity depends only on transport of
electrons and our results are in good agreement with those
in ref [11] at one atmosphere.
5.2. Thermal conductivity
We have considered only translational thermal
conductivity of heavy species λ tr and electrons λ tr ,
than reactive and internal thermal conductivity have not
been taken into account in the present calculation. These
two coefficients were calculated by using the third
approximation of the Chapman-Enskog method for the
electron component Eq. (12) and the first approximation
for heavy components Eq. (11):
h
Fig. 1. Mole fractions of different species in helium
plasmas at different pressures.
5. Transport Coefficients of Multicomponent Mixture
Viscosity, thermal conductivity and electrical
conductivity of thermal helium plasma are computed on
basis of the pre-calculated equilibrium plasma
compositions.
5.1. Electrical conductivity
Devoto [10] has developed analytical expressions of
electrical conductivity for partially ionized gas as function
of collision transport cross-section. We have calculated
the electrical conductivity of helium plasma using the
third approximation of Devoto method, and the
expression is given by:
𝜎=
3
𝑒 2 𝑛𝑒2
2
�
2𝜋
𝑚𝑒 𝐾𝐾
�
1�
2
𝑞 11 𝑞 12
�
� 12
𝑞
𝑞 22
𝑞 00 𝑞 01 𝑞 02
�𝑞 01 𝑞 11 𝑞 12 �
𝑞 02 𝑞 12 𝑞 22
(10)
λtrh
L11

Lv1
x
= 1
L11

Lv1
 L1v x1

 
 Lvv xv

xv 0
 L1v


 Lvv
 2πk T 
75
λ = ne k B  B e 
8
 me 
e
tr
e
1
2
(11)
q 22
( )
q 11 q 22 − q 12
2
(12)
The element of matrix L ij can be found in ref [2] which
are function of collision integrals Eq.(7) and the elements
x i correspond to the mole fraction of each species (He,
He+, He++ and e-). The elements qij are given in ref [9].
Fig. 3 shows the temperature dependence of the
translational thermal and electron thermal conductivity in
helium plasma for different pressures.
where the elements of determinant qij are given in [10]
and they are function of collision cross-section. We can
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3
Fig. 4. Evolution of Viscosity of partially helium plasma
with temperature for different pressures.
Fig. 3.
Evolution of heavy translational thermal
conductivity (top) and electron thermal conductivity
(bottom) of partially helium plasma with temperature for
different pressures.
5.3. Viscosity
The contribution of electron to the viscosity is
negligible than only heavy species contribute to the total
viscosity. Chapman-Enskog method have been used in
the first approximation to evaluate the viscosity:



H1v x1
 
H vv xv
x
 xv 0
η =− 1
H11  H1v



H v1  H vv
H11

H v1
(13)
7. References
[1] M. Vardelle. J. Thermal Spray Technol., 10, 2
(2001)
[2] J.O. Hirschfelder. Molecular theory of gases and
liquids. (Wiley) (1954)
[3] S. Chapman and T.G. Cowling. The Mathematical
Theory of Non-uniform Gases.
(Cambridge:
Cambridge University Press) (1970)
[4] J.C. Rainwater. J. Chem. Phys., 77, 1 (1982)
[5] G.B. Benson.
Massachusetts Institute of
Technology. (1996)
[6] L.D. Higgins and F.J. Smith. Molecular Phys., 14,
4 (1968)
[7] A. Mahfouf, P. André and G. Faure. in: 9th Int.
Student Conf. on Advanced Science and Technology.
(Clermont-Ferrand) 27 (2014)
[8] M. J. Slaman and R.A. Aziz. Int. J. Thermophys.,
12, 5 (1991)
[9] F. Bendjebbar and P. André. Plasma Sources Sci.
Technol., 14, 8 (2012)
[10] R.S. Devoto. Phys. Fluids, 10, 2105 (1967)
[11] R.S. Devoto. J. Plasma Phys., 2, 1 (1968)
[12] P. André, G. Faure, S. Lalléchère and A. Mahfouf.
in: 17th Int. Symp. on Electro Magnetic
Compatibility. (Clermont-Ferrand) 18 (2014)
The element of matrix H ij can be found in ref [2] which
are function of collision integrals Eq.(7). Fig. 4 shows the
temperature dependence of the viscosity in helium plasma
for different pressures.
6. Conclusion
Viscosity, thermal conductivity, and electrical
conductivity were computed successfully for the
equilibrium helium plasma, and the results can be used as
input data for the reliable numerical simulation of thermal
plasmas.
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