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Thermodynamic and transport properties of two-temperature lithium plasmas
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2012 J. Phys. D: Appl. Phys. 45 165202
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IOP PUBLISHING
JOURNAL OF PHYSICS D: APPLIED PHYSICS
J. Phys. D: Appl. Phys. 45 (2012) 165202 (11pp)
doi:10.1088/0022-3727/45/16/165202
Thermodynamic and transport properties
of two-temperature lithium plasmas
Hai-Xing Wang1,3 , Shi-Qiang Chen1 and Xi Chen2
1
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100191,
People’s Republic of China
2
Department of Engineering Mechanics, Tsinghua University, Beijing 100084,
People’s Republic of China
E-mail: [email protected]
Received 21 December 2011, in final form 15 February 2012
Published 30 March 2012
Online at stacks.iop.org/JPhysD/45/165202
Abstract
Thermodynamic and transport properties of two-temperature lithium plasmas are calculated
for the pressure range from 0.0001 to 1 atm, electron temperatures from 1000 to 40 000 K and
the electron/heavy particle ratio from 1 to 5. The calculated results are presented concerning
the variation with the electron temperature of the plasma composition, specific enthalpy,
specific heat, viscosity, electrical conductivity, thermal conductivity and diffusion coefficient
with the gas pressure and electron/heavy-particle temperature ratio as the parameters. The
effects of gas pressure and electron/heavy-particle temperature ratio on lithium plasma
properties are discussed in terms of the variation of gas ionization degree with gas pressure and
temperature. For the special case with gas temperature below 10 000 K and without accounting
for gas ionization, the present calculated results about lithium viscosity and thermal
conductivity at atmospheric pressure are consistent with those reported in the literature.
(Some figures may appear in colour only in the online journal)
of the cathode material would result in longer lifetime of the
lithium thruster. Another advantage of using lithium as the
propellant is higher thrust efficiency. It has been reported that
the thrust efficiency of a lithium thruster may be as high as
60% or higher. It is believed that the higher thrust efficiency
is mainly due to the special feature of the lithium propellant,
i.e. its lower first ionization potential (5.4 eV) and much higher
second ionization potential (75.6 eV). This special feature can
reduce the ionization loss and the frozen flow loss in the thruster
nozzle [2, 3].
The MPDT is a hybrid accelerator in which both
electromagnetic and gas dynamic processes contribute
significantly to the acceleration of the propellant (plasma).
The gas dynamic acceleration is strongly coupled with the
electromagnetic fields due to the presence of Joule heating and
Lorentz force, and many other complicated factors are also
involved in the thruster, such as the electrode phenomenon,
plasma non-equilibrium, conjugate gas–solid heat transfer and
rarefied gas effect. Although experimental techniques can
provide useful data about the thruster, many quantities of
interest are not easily obtained in some important regions of
the thruster, where small space size, harsh environment of high
1. Introduction
Gaseous lithium is often considered as a favourite propellant
for a magnetoplasmadynamic thruster (MPDT) since it can
give longer service lifetime (>1000 h), and higher performance
and efficiency for the thruster [1]. The effective work function
of the cathode can be appreciably lowered when lithium is used
as the working fluid of the thruster—the effective work function
of the cathode is 4.5 eV for a tungsten–nitrogen discharge,
while it is reduced to about 2.9 eV for a tungsten–lithium
system. Additionally, when lithium is used as the working fluid
of the thruster, a multi-channel discharge would occur at the
cathode surface with lower current density at the arc roots for a
given total current. Hence, lower cathode surface temperatures
and thus substantially lower erosion rate or evaporation loss of
the cathode material would be achieved. Since the evaporation
loss of the cathode material has been demonstrated to be the
major erosion mechanism of thruster cathode in a steady-state
MPDT and directly affects the service lifetime of the thruster,
lower cathode surface temperatures and lower evaporation loss
3
Author to whom any correspondence should be addressed.
0022-3727/12/165202+11$33.00
1
© 2012 IOP Publishing Ltd
Printed in the UK & the USA
J. Phys. D: Appl. Phys. 45 (2012) 165202
H-X Wang et al
temperature and/or condensing lithium vapour would make
the optical and/or probe measurements extremely difficult.
Numerical simulation or modelling has thus become a powerful
auxiliary tool to understand the internal processes of the
thruster. In addition, for deep-space exploration missions, the
MPDT usually operates at a power level of several megawatts
(MW) but ground testing of these electric thrust devices at
MW-class power levels presents formidable technological and
economic challenge. For this case numerical simulation may
also play an important role in understanding and predicting
the performance of the thrusters operating at different power
levels to optimize the thruster and its electrode geometry.
The numerical simulation of MPDT or other types
of electric thrusters usually involves the solution of the
mass, momentum and energy equations of the working fluid
(lithium plasma) and the solution of the electromagnetic
field equations strongly coupled with the fluid dynamical
equations. Thermodynamic and transport properties of the
working fluid (plasma) appear in the governing equations, and
thus need to be known in advance. Under electric thruster
conditions, plasma properties are the function of gas pressure,
electron temperature and heavy-particle temperature. Hence,
obtaining reliable thermodynamic and transport properties
and compiling databases of plasma properties for different
pressures and different electron and heavy-particle temperature
ratios are necessary for any successful numerical simulation.
The calculation of the composition and thermodynamic
properties of plasmas (such as the specific enthalpy, specific
heat and mass density) is relatively simpler since the
fundamental data required for the calculation are well
known [4]. The Chapman–Enskog method, which has been
successfully used in calculating the transport coefficients of
non-ionized gases [5, 6], has been extended to the calculation of
the transport coefficients of plasmas, including their viscosity,
thermal conductivity, electrical conductivity and diffusion
coefficients [7–23]. Some uncertainties exist in the calculated
values of the transport coefficients, and this is mainly due to
the uncertainties in the values of the intermolecular potentials
and collision integral values used in the calculation of transport
coefficients.
Many calculated results of transport properties of plasmas
are available in the literature. For the case of plasmas in
local thermodynamic equilibrium (LTE) and local chemical
equilibrium (LCE) states, calculated results of the transport
properties of many types of thermal plasmas are available (e.g.
see [7–13]). The involved plasma-forming gases include Ar
[7–9], N2 [9], O2 [9], air [10], He [11, 12], H2 [13], Ar–
H2 [13], Ar–O2 [9], Ar–N2 [9], Ar–air [10], air–O2 [10],
air–N2 [10], Ar–He [12] and so on. There are also many
papers [14–25] concerning the calculation of the properties of
two-temperature (2-T) plasmas such as Ar [14–17], He [18],
Ar–H2 [19], Ar–He [20], nitrogen–hydrogen [19], carbon–
oxygen [21], nitrogen–oxygen [22] and oxygen [17, 23]. Since
the electron mass is considerably smaller than the heavyparticle mass, the coupling effects between electrons and heavy
particles in the transport processes are expected to be weak,
and thus a simplified theory is usually used in the calculation
of transport coefficients [7, 8], in which the coupling between
electrons and heavy particles is completely ignored. Recently,
Rat et al [24] have developed a theory in which the coupling
between heavy particles and electrons is not neglected, giving
rise to a new set of transport coefficients and coupling terms in
the flux definition. Rat et al compared the results obtained
by their non-simplified approach and those obtained from
the simplified theory of transport properties of Bonnefoi and
showed that [16, 25] there exist non-negligible discrepancies
between them, especially at high temperatures, for electrical
conductivity and electron translational thermal conductivity,
while the simplified approach is correct for viscosity, which
depends mainly on the heavy species and very little on the
electrons. A recent study [17, 19, 21] of non-LTE transport
properties based on a comparison of the approaches of Devoto
and Rat et al shows that coupling between electrons and heavy
species does not lead to significant changes in the predicted
non-equilibrium plasma transport properties, except for certain
ordinary diffusion coefficients. No significant discrepancies
occur in the total thermal conductivity (including the reactive
contribution), viscosity or electrical conductivity, even when
they depend on the ordinary diffusion coefficients.
Although a lithium plasma is preferred to be used as
the propellant of the MPDT, [26, 27] addressed only its
atmospheric-pressure transport properties for gas temperatures
lower than 10 000 K, and gas ionization was not considered
in these studies. So far no complete sets of thermodynamic
and transport properties are available in the literature for
a lithium plasma. This paper intends to fill this gap.
Thermodynamic and transport properties of the LTE and/or
2-T lithium plasma are calculated for the pressure range from
0.0001 to 1 atm, electron temperatures from 1000 to 40 000 K
and the electron/heavy particle ratio from 1 to 5.
2. Calculation of plasma composition and
thermodynamic properties
The prerequisite for calculating the thermodynamic properties
and transport coefficients of a plasma is that the plasma
composition is known.
In the following composition
calculation, the lithium plasma is assumed to consist of four
species, i.e. lithium atoms (Li), singly ionized lithium ions
(Li+ ), doubly ionized lithium ions (Li2+ ) and electrons (e).
Maxwellian velocity distribution function (MVDF) with the
characteristic temperature Th is assumed to be applicable to
heavy species, whereas the MVDF for electrons is with the
characteristic temperature Te . For the 2-T plasma the electron
temperature (Te ) may be equal or appreciably higher than
the heavy-particle temperature (Th ), especially at low gas
pressures. For a 2-T plasma with given pressure, heavyparticle temperature (Th ) and electron temperature (Te ), the
composition of the lithium plasma can be calculated by
simultaneously solving the Saha equations for single and
double ionization of lithium, the equation of state and the
equation of electric quasi-neutrality, i.e.
ne ns
2π me kB Te 3/2 Qs,el (Te )
−Ea
= 2 exp
na
kB T e
h2
Qa,el (Te )
2
(1)
J. Phys. D: Appl. Phys. 45 (2012) 165202
H-X Wang et al
ne nd
2πme kB Te 3/2 Qd,el (Te )
−Es
(2)
= 2 exp
ns
kB T e
h2
Qs,el (Te )
ni
(3)
p = kB Te ne + kB Th
by the species in the plasma is assumed to be a first-order
perturbation to the Maxwellian distribution. The perturbation
is then expressed in a series of Sonine polynomials [5, 6],
which finally through linearization of the Boltzmann equation
leads to a system of linear equations that can be solved for
different transported quantities to obtain different transport
coefficients. While the elements in the matrix of the system
of equations depend on the interaction between associated
colliding species in terms of collision integral values, the
number of elements depends on the order of the chosen
approximation. In this work, thermal conductivity, electrical
conductivity and viscosity are calculated by the simplified
approach of Devoto [7, 8], while the calculation of diffusion
coefficients is performed following the non-simplified method
of Rat et al [24, 25] according to the new finding in the study
of Colombo et al [17, 19, 21].
j,j =e
nj Zj = 0
(4)
j
where nj and mj are the particle number density and the
particle mass of the j th species, with subscript j = a, d,
s and e or 1, 2, 3 and 4 to denote Li atom, doubly ionized
ion Li2+ , singly ionized ion Li+ and electron e, respectively.
p is the total pressure of the plasma system, Zj is the
electric charge number of the j th particle, kB and h are the
Boltzmann constant and Planck constant. Qa,el (Te ), Qs,el (Te )
and Qd,el (Te ), respectively, represent the partition functions of
the atoms, singly ionized ions and doubly ionized ions, and
they can be evaluated from the energy levels of the electronic
states, e.g. from those tabulated by Moore [4]. In this study,
it is assumed that the populations of excited levels inside
the atoms and ions are governed by the electron temperature
(Te ). Ea and Es are the ionization potential for the single and
double ionization of the plasma, and their values are 5.4 eV and
75.6 eV, respectively, for the lithium plasma. Equations (1)
and (2) are the well-known mass action law for the LTE
plasmas, and [28] showed that they are also applicable to the
2-T plasmas.
The electron/heavy-particle temperature ratio Te /Th will
be expressed by the parameter θ (θ = Te /Th ), which represents
the degree of thermal non-equilibrium of the plasma. For a
given gas pressure, one can calculate the number densities
of each species of the lithium plasma as functions of the
electron temperature with θ as a parameter by solving the set
of equations (1)–(4).
After the plasma composition is calculated, the mass
density of the plasma can be calculated by
ρ=
nj mj ≈ (na + ns + nd )ma .
(5)
3.1. Collision integral values
Collision integral values are required to calculate the transport
coefficients. Accounting for the interaction between colliding
species i and j , the collision integral is defined as
= 2π kB Tij∗ /m∗ij
(l,s)
ij
∞ ∞
×
exp(−γij2 )γij2s+3 (1 − cosl χ )b db · dγij
(8)
0
0
where m∗ij and Tij∗ are, respectively, the reduced mass
and effective temperature of colliding particles, which are
expressed as
m∗ij = mi mj /(mi + mj ),
Tij∗ = [(mi /Tj + mj /Ti )/(mi + mj )]−1 .
(9)
In equation (8), b and χ are the impact parameter and reflection
angle, respectively. Ti and Tj are the temperatures of species
i and j , respectively. γij is a function of the relative velocity
gij between species iand j , which is given by [5]
γij = gij / 2kB Tij∗ /m∗ij .
(10)
j
The specific enthalpy (J kg−1 ) of the plasma is calculated by
∂ ln Qj,int
1
kB 5 kB nj Tj +
nj E j +
nj Tj2
h=
ρ j
ρ j
2 ρ j
∂Tj
The deflection angle, χ , is associated with the interaction
potential V (r) as follows [5]:
∞
χ = π − 2b
dr/(r 2 1 − 2V (r)/m∗ij gij2 − b2 /r 2 ) (11)
(6)
rm
where ρ is the mass density of the plasma, Ej represents the
ionization energy of the j th species, whereas Qj,int is the
internal partition function of the j th particle. The specific
heat at constant pressure of the plasma can then be calculated
from the specific enthalpy by
∂h cp =
.
(7)
∂Te p.θ
where r is the separate distance between the interacting
particles.
The values of the collision cross sections are obtained from
the collision integral values using the relationship [5]
l
∗
∗
= 4(l +1)(l,s)
Q(l,s)
ij
ij /{ kB Tij /2π mij (s +1)![2l +1−(−1) ]}.
(12)
Collision integral values for the interactions between pairs
of species present in the gases and gas mixtures of interest
were obtained in different ways. Tabulations of some collision
integral values have been published in the literature and these
values could be used directly in the calculation of transport
coefficients. In some cases, the collision integral values are
3. Calculation of the transport properties of 2-T
plasmas
The transport coefficients are calculated using the well-known
Chapman–Enskog method. The distribution function followed
3
J. Phys. D: Appl. Phys. 45 (2012) 165202
H-X Wang et al
mp
calculated from the published interaction potentials V (r) using
equation (8) as well as equation (12) [29, 30].
Holland et al [26] showed that the collision integral
values for Li–Li interactions can be calculated accurately
by the Hulburt–Hirschfelder (H–H) potential. The excellent
agreement of calculated viscosity with experimental results
within a low temperature range provides evidence that H–H
potential can be used to accurately estimate the transport
properties. So, the collision integral values for Li atom
interaction derived from Holland [26] are used in our
calculation.
Charge-exchange cross sections are important for a
partially ionized gas, since they govern the rate of diffusion
of the ions through the neutrals. However, in the temperature
ranges of interest there are no available information reported
concerning experimentally measured results of the Li–Li+
charge-exchange cross sections. In this study, the used chargeexchange cross sections of Li–Li+ are obtained by a scaling
procedure based on experimental measurements for caesium
in [31]. The elastic collision integrals of Li–Li+ used here are
also obtained from [31], while the cross section of electron–Li
is obtained from [32].
The collision integral values for interactions between
charged species are calculated by considering the screened
Coulomb potential [7, 33, 34],
ϕ(r) = ϕ0 (λD /r) exp(−r/λD )
The elements qij appearing in equations (15) and (16) are
calculated as in [37].
The reactive thermal conductivity of electrons and heavy
species are, respectively, given by [38, 39]


2
3
∂p
T
n
∂p
j
h
e
a
a
ke,r =
hr 
mj Drj
+ me Dre
ρk
T
∂T
T
∂T
B
h
e
e
e
r=1
j =1
(17)
kh,r =
2
r=1


3
∂p
n
∂p
T
j
h
e
a
a

hr 
mj Drj
+ me Dre
ρkB Th
∂T
T
∂T
h
e
h
j =1
(18)
a
in which Drj
is the ambipolar diffusion coefficient [38, 39],
and j stands for the heavy species including atoms, singly and
doubly ionized ions. The total thermal conductivity of the 2-T
plasma system is thus given by
k = ke,tr + kh,tr + kr = ke,tr + kh,tr + kh,r + ke,r .
3.2.2. Electrical conductivity. As shown in [16], the
electrical conductivity σ of a 2-T plasma can be written as


4 
4

2 e n
Zi
σ =−
mj nj Zj Dij .
(20)
 Ti

ρkB
(13)
The contribution of heavy species can be neglected and the
electrical conductivity is reduced to a good approximation to
σ ≈
where ε0 and ne are, respectively, the vacuum permittivity and
electron number density.
3.2.1. Thermal conductivity. For a 2-T plasma, the thermal
conductivity consists of translational thermal conductivity
of electrons, translational thermal conductivity of heavy
species and reactive thermal conductivity. As shown by
Devoto [35–37], the third approximation of translational
thermal conductivities of electrons and fourth approximation
of translational thermal conductivities of heavy species are,
respectively, given by
1
75n2e kB 2πkB Te
(15)
ke,tr =
8
me q 11 − (q 12 )2 /q 22
kh,tr
qij02
qij12
qij22
qij32
0
qij03
qij13
qij23
qij33
0
3
e2 n (mj nj Zj Dj 4 ).
ρkB Te j =1
(21)
3.2.3. Viscosity. The viscosity is computed using the second
approximation in the Sonine polynomial expansion and is
given by
3.2. Transport coefficients
qij01
qij11
qij21
qij31
√
nj / mj
j =1
i=1
where ϕ0 = Zi Zj e2 /4πε0 λD . In equation (13), λD is the
Debye length associated with electrons and is given by [7, 19]
ε0 k B T e
λD =
(14)
e 2 ne
00
qij
10
√
q
75kB 2πkB Th ij20
=−
qij
8
|q|
q 30
ij
0
(19)
µ=
5(2π kB Th )1/2
2|q|
00
q
ij
10
q
ij
nj
01
q ij
11
q ij
0
1/2
ni mi 0 .
0 (22)
The elements appearing in this matrix are also calculated in
terms of the expression presented in [37].
3.2.4. Diffusion coefficient. Diffusion coefficient calculation
starts from binary diffusion coefficients and then the ordinary
and ambipolar diffusion coefficients are computed in terms
of them. The binary diffusion coefficient is evaluated based
on the derivation of two-temperature transport properties, as
reported by Rat et al [16, 19]:
0
ni 0 .
0 0
(16)
Dijb
4
=
2π kB
m∗ij Tij∗
1/2
3Ti gij
16nij(1,1)
(23)
J. Phys. D: Appl. Phys. 45 (2012) 165202
H-X Wang et al
where n is the total number density, ij(1,1) , Ti and Tj are
the collision integral and the temperatures of species i and
j , respectively. θij = Ti /Tj , and
gij =
(mj + mi θj i )2 (mi + mj θj i )3
5/2
θj i (mj + mi )5
.
(24)
Once the temperatures are known, six dependent binary
diffusion coefficients can be calculated for the present lithium
plasma system using already available ij(1,1) values, and all
other diffusion coefficients derived from these binary diffusion
coefficients can be computed employing standard relationships
[6]. The first-order approximation to the ordinary diffusion
coefficients can be written as
Dij = (F j i − F ii )/(mj |F |)
where F ij is defined as follows [6]:


ni m j 
1 ni
(1 − δij ).
Fij =  b +
ρ Dij l=i mi Dilb
(25)
(26)
Here ρ, δij , ni and mi represent the mass density, Kronecker
delta symbol, number density and mass of species i,
respectively.
4. Results
In this section typical calculated results are presented
concerning the variations with electron temperature of the
thermodynamic and transport properties of the 2-T lithium
plasma, taking the gas pressure p and/or the electron/heavyparticle temperature ratio θ (= Te /Th. ) as the parameters. In
the calculation, the electron temperature is taken to vary from
1000 to 40 000 K, the pressure varies from 0.0001 to 1 atm,
while the values of temperature ratio θ are chosen to be 1, 2
and 5. The selected temperature and pressure ranges are close
to those encountered in the electric propulsion devices.
Figure 1 presents the variation of the lithium plasma
composition with electron temperature for three different
electron/heavy-particle temperature ratios. The gas pressure is
fixed to be 0.0001 atm. It is seen that since the first ionization
potential of lithium atoms is comparatively low (5.4 eV),
significant first ionization of lithium atoms occurs at lower
electron temperatures (around 4000 K). On the other hand, the
second ionization potential of lithium is much higher (75.6 eV);
the singly ionized lithium ions undergo their notable double
ionization only when the electron temperature is higher than
about 30 000 K. Within the wide range of electron temperatures
from about 4000 to about 30 000 K, the molar fractions of
singly ionized lithium ions and electrons are almost unchanged.
Since the second ionization degree of the lithium propellant is
very low as the electron temperature is lower than 30 000 K, it
is expected that the change in the ionization degree within the
thruster nozzle would be small and thus the frozen flow loss
in the nozzle would be significantly reduced for the lithium
electric-propulsion device.
Figure 1. Mole fraction of different species in 2-T lithium plasmas
for three different values of the electron/heavy-particle temperature
ratio θ: (a) θ = 1, (b) θ = 2, (c) θ = 5. The gas pressure is fixed to
be at 0.0001 atm.
Variations of the total specific enthalpy, heavy-species
specific enthalpy and electron specific enthalpy of the 2-T
lithium plasma with electron temperature are presented in
figure 2 for three different electron/heavy-particle temperature
ratios and three different pressures. It is seen that the
specific enthalpy changes almost linearly with Te within
a wide electronic temperature range from about 10 000 to
about 30 000 K, corresponding to the fact that the plasma
5
J. Phys. D: Appl. Phys. 45 (2012) 165202
H-X Wang et al
composition is almost unchanged in this wide temperature
range. Deviation from this linear change occurring at lower
electron temperatures (lower than about 10 000 K) is due to
the plasma composition change caused by the first ionization
of lithium atoms, while deviation from this linear change
occurring at significantly higher electron temperatures (higher
than about 30 000 K) is due to the plasma composition change
caused by the second ionization of lithium. It is also seen
that in this high-temperature region, the specific enthalpy is
much higher at a low gas pressure (e.g. 0.0001 atm) and a
low temperature ratio (e.g. θ = 1) since more notable second
ionization is involved. With the increase in the temperature
ratio θ or gas pressure the specific enthalpy curves shift towards
higher Te , and the electron temperature at which the single or
double ionization becomes significant also increases.
Corresponding to the calculated results of specific
enthalpy, the calculated variations of the total specific heat,
heavy-species specific heat and electron specific heat with
electron temperature at a constant pressure of the 2-T lithium
plasmas are shown in figure 3 for three different temperature
ratios and three different pressures. It is seen that the
specific heat is almost independent of the electron temperature
within a wide electronic temperature range from about 10 000
to about 30 000 K, and the values of the specific heat are
comparatively small for this electron temperature range with
almost constant molar fraction of the singly ionized lithium
ions. The specific heat assumes peak values in the temperature
range with significant change in the single ionization degree
(for Te less than about 10 000 K) or with significant change in
the double ionization degree (Te higher than about 30 000 K).
In the temperature range with a significant change in the gas
ionization degree, the values of the specific heat appreciably
depend on the gas pressure and temperature ratio.
The calculated variations of the lithium plasma viscosity
with electron temperature are presented in figure 4 for three
different temperature ratios and three different pressures.
Since the viscosity is associated with the amount of momentum
transfer among gas species, the viscosity would be caused
mainly by heavy particles while the contribution due to
electrons is always small. It is seen from figure 4 that the peak
values of lithium plasma viscosity appear in the temperature
range where an appreciable change in the ionization degree
of lithium atoms appears. For the range with electron
temperatures higher than about 10 000 K, the singly ionized
lithium ions dominate the momentum transfer while the molar
fraction of the singly ionized ions depends less on the electron
temperature, as shown in figure 1. Hence, the plasma viscosity
is also less dependent on the electron temperature. Within
this temperature range, the viscosity increases with a decrease
in temperature ratio θ . The viscosity also increases with
increasing pressure, since more heavy-particle number density
would be involved.
For three different temperature ratios and three different
pressures, the calculated variations of the electrical
conductivity of the 2-T lithium plasma with electron
temperature are shown in figure 5. Since the plasma
electrical conductivity is associated mainly with electrons
with high mobility, its values at low gas temperatures are
Figure 2. Variations of the total specific enthalpy (a), specific
enthalpy of heavy species (b) and specific enthalpy of electrons
(c) of 2-T lithium plasmas with electron temperature for three
different pressures and three different temperature ratios.
6
J. Phys. D: Appl. Phys. 45 (2012) 165202
H-X Wang et al
Viscosity, kg m-1s-1
10-4
10
-5
10
-6
10
-7
10
-8
p=1 atm
p=0.01 atm
p=0.0001 atm
θ =1
θ =2
θ =5
10-9
Electron temperature,10 4K
3
Electrical conductivity,10 Sm
-1
Figure 4. Variations of the viscosity of 2-T lithium plasmas with
electron temperature for three different pressures and three different
temperature ratios.
25
20
15
p=1 atm
p=0.01 atm
p=0.0001 atm
θ=1
θ=2
θ=5
10
5
0
Electron temperature,10 4K
Figure 5. Variations of the electrical conductivity of 2-T lithium
plasmas with electron temperature for three different pressures and
three different temperature ratios.
very small and can be ignored. On the other hand, figure 5
shows that in the region with appreciable gas ionization,
the electrical conductivity increases rapidly with increasing
electron temperature. For a constant electron temperature, the
electrical conductivity increases significantly with increasing
gas pressure due to the larger electron number density involved
at higher pressures. The electrical conductivity somewhat
increases with increasing temperature ratio. It is seen that
at a pressure of 0.0001 atm, there appears a valley in the
electrical conductivity curve when the electron temperature is
higher (around 35 000–40 000 K). This decrease in electrical
conductivity can be attributed to the non-negligible presence of
doubly charged ions Li2+ , which tend to increase the Coulomb
collision integrals. The Coulomb cross-section for interactions
between doubly charged ions and electrons is much larger than
that for interactions between singly charged ions and electrons,
Figure 3. Variations of the total specific heat (a), specific heat of
heavy species (b) and specific heat of electrons (c) of 2-T lithium
plasmas with electron temperature at constant pressure for three
different pressures and three different temperature ratios.
7
H-X Wang et al
-1
J. Phys. D: Appl. Phys. 45 (2012) 165202
Total thermal conductivity, Wm K
Total
Heavy species
Reaction
Electron
1.5
1
0.5
0
0.5
1
1.5
p=1 atm
p=0.01 atm
p=0.0001 atm
θ=1
θ=2
θ=5
12
-1
Total thermal conductivity, Wm-1K-1
2
10
8
6
4
2
0
2
4
Electron temperature, 10 K
4
Electron temperature,10 K
Figure 7. Variations of the total thermal conductivity of 2-T lithium
plasmas with electron temperature for three different pressures and
three different temperature ratios.
Figure 6. Variations of the total thermal conductivity of 2-T lithium
plasmas with electron temperature for the special case with pressure
0.01 atm and temperature ratio θ = 2.
12
conductivity, Wm-1K-1
Electron translational thermal
so the electrical conductivity decreases. This behaviour was
also observed in the calculation of electrical conductivity of
argon [16] and nitrogen-oxygen plasmas [22].
Figure 6 shows a typical calculated result (for 0.01 atm
and θ = 2) concerning the variations of the total thermal
conductivity of the lithium plasma with electron temperature
and its components due to the translational energy transport
of electrons and heavy particles and due to the energy
transport caused by the ionization–recombination reaction
(reactive thermal conductivity). It is seen from figure 6
that in the region with appreciable variation in the ionization
degree of lithium atoms, reactive thermal conductivity is the
dominant component, while translational thermal conductivity
of electrons is the dominant component of the total thermal
conductivity at high electron temperatures (higher than
∼8000 K).
The effects of gas pressure and the temperature ratio θ
on the total thermal conductivity are shown in figure 7. It is
seen that the gas pressure significantly affects the total thermal
conductivity with higher values of thermal conductivity at
higher pressures. The temperature ratio also somewhat affects
the total thermal conductivity. The effects of gas pressure
and the temperature ratio θ on the electron translational
thermal conductivity and electron reactive conductivity are
shown in figure 8. It is seen that the electron translational
thermal conductivity strongly depends on the gas pressure
with larger values at higher pressures (associated with higher
electron number densities). The temperature ratio θ also
affects the electron translational thermal conductivity, but only
slightly. The peaks of electron reactive thermal conductivity
are observed in the region with an appreciable change in first
and second ionization, as shown in figure 8(b). Figure 9 shows
the effects of gas pressure and the temperature ratio θ on the
heavy particles’ translational thermal conductivity and reactive
thermal conductivity. It is seen that for a fixed gas pressure
and a given temperature ratio, the heavy-particle translational
thermal conductivity assumes greater values at lower electron
p=1 atm
p=0.01 atm
p=0.0001 atm
=1
=2
=5
10
8
6
4
2
0
Electron temperature, 10 4K
(a) translational thermal conductivity of electrons
conductivity, Wm K
-1
0.8
-1
Electron reactive thermal
1
0.6
p=1 atm
p=0.01 atm
p=0.0001 atm
=1
=2
=5
0.4
0.2
0
4
Electron temperature, 10 K
Figure 8. Variations of the translational thermal conductivity of
electrons (a) and reactive thermal conductivity of electrons (b) of
2-T lithium plasmas with electron temperature for three different
pressures and three different temperature ratios.
8
thermal conductivity, Wm-1K-1
Heavy species translational
J. Phys. D: Appl. Phys. 45 (2012) 165202
10-1
10-2
H-X Wang et al
p=1 atm
p=0.01 atm
p=0.0001 atm
=1
=2
=5
10-3
10-4
10-5
Electron temperature, 10 4K
0.1
conductivity, Wm-1K-1
Heavy species reactive thermal
(a) translational thermal conductivity of heavy species
0.08
0.06
p=1 atm
p=0.01 atm
p=0.0001 atm
=1
=2
=5
Figure 10. Variations of the ordinary diffusion coefficient of De,Li
of 2-T lithium plasmas with electron temperature for three different
pressures and three different temperature ratios.
the transport properties of lithium vapour with temperatures
below 10 000 K. Additionally, the vapour was assumed to
consist entirely of atoms, and thus atom ionization and
electronic excitation at higher temperatures were ignored.
However, ionization may influence the lithium properties even
at considerably low temperatures (e.g. 4000–10 000 K), since
lithium has a lower first ionization potential. For the case
of non-ionized lithium and at atmospheric pressure, figure 11
compares our calculated viscosity with the results reported
previously by Holland [26] for non-ionized lithium vapour. It
is seen that the viscosity reported in [26] is almost linear with
Te for lithium vapour temperatures up to 10 000 K. Actually,
as shown in figure 1, the first ionization of lithium starts at
around 4000 K. Once the lithium ionization starts, the viscosity
will decrease with the increase in temperature. Since the
effects of ionization of lithium vapour are not included in
Holland’s study, he overestimated the lithium viscosity at high
temperatures. When gas ionization is ignored, our calculated
results are consistent with those reported by Holland [26].
However, the behaviour of thermal conductivity cannot be
predicted accurately without considering the effect of gas
ionization. A similar comparison concerning the thermal
conductivity of lithium previously reported by Holland [26]
with our predicted results is given in figure 12. It is also seen
that when gas ionization is ignored, our predicted results are
consistent with those reported by Holland [26]. For this case,
the calculated thermal conductivity increases linearly with
gas temperature for lithium gas temperatures below 10 000 K.
However, when gas ionization is included in the calculation,
the reactive thermal conductivity would become important, and
Holland’s calculation appreciably underestimated the thermal
conductivity in the temperature range with appreciable gas
ionization. The behaviour of thermal conductivity also cannot
be predicted accurately without considering the effect of gas
ionization.
0.04
0.02
0
Electron temperature, 104K
Figure 9. Variations of the translational thermal conductivity of
heavy species (a) and reactive thermal conductivity of heavy species
(b) of 2-T lithium plasmas with electron temperature for three
different pressures and three different temperature ratios.
temperatures with an appreciable change in the atom ionization
degree. The maximum value of the heavy-particle translational
thermal conductivity is larger at higher gas pressures or at lower
electron/heavy-particle temperature ratios.
The diffusion coefficients between different plasma
species depend on the gas pressure, electron temperature
and electron/heavy-particle temperature ratio. As a sample
calculated result, for three different gas pressures and three
different temperature ratios, the calculated variations of the
ordinary diffusion coefficient between electrons and lithium
atoms, i.e. De−Li , with electron temperature are presented in
figure 10. It is seen that the diffusion coefficient is significantly
influenced by the gas pressure with much higher values at lower
pressures. The temperature ratio also appreciably affects the
diffusion coefficient with higher values at smaller temperature
ratios.
Although lithium has been considered as a preferable
propellant of the electrical propulsion system for a long time,
the thermodynamic and transport properties of lithium plasmas
at high temperatures are not available. Previously reported
calculations of lithium properties were only concerned with
9
J. Phys. D: Appl. Phys. 45 (2012) 165202
8
viscosity and thermal conductivity agree well with those
previously reported in the literature. However, gas ionization
significantly affects the calculated results of plasma properties.
The properties are also influenced by the electron/heavyparticle temperature ratio for the two-temperature lithium
plasma.
6
Acknowledgments
4
The authors wish to thank Professor Edgar Y Choueiri of
EPPDyL of Princeton University for helpful suggestions and
discussions. This work was supported by the National
Natural Science Foundation of China (Grant Nos 50836007
and 11072020).
-1 -1
12
10
Holland, metal vapor
Our result, metal vapor
Our result, plasma
-5
Viscosity, 10 kg m s
H-X Wang et al
2
0
0.2
0.4
0.6
0.8
4
Lithium temperature, 10 K
1
References
2
-1
Thermal conductivity, Wm K
-1
Figure 11. Comparison of the calculated values of lithium viscosity
obtained from this study and those previously reported by Holland
at 1.0 atm and θ = 1.
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5. Conclusions
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11