CHIN. PHYS. LETT. Vol. 30, No. 11 (2013) 114401
Simulation of Radiative Transfer in Nonequilibrium Plasmas Containing N and O
Species Based on the Approximate Collision-Radiative Method *
HE Xin(何新)** , CHANG Sheng-Li(常胜利), DAI Sui-An(戴穗安), YANG Jun-Cai(杨俊才)
Department of Physics, National University of Defense Technology, Changsha 410073
(Received 24 April 2013)
An approximate collision-radiative method is developed to calculate energy level populations of atoms and applied to simulate radiative transfer in a shock layer. Considering nonequilibrium plasmas with the electronic
temperature below 20000 K, we introduce correction factors to the Saha–Boltzmann equation and simplify the
master equation of the full collision-radiative model. Based on the method, the error of heat flux is smaller than
5% in a two-cell problem and smaller than 7% in the Stardust re-entry flow field compared with the correlated-𝑘
method. Furthermore, almost a factor of 2 is obtained in the improvement of computational efficiency compared
with the nonequilibrium air radiation (NEQAIR) code. The results show that the present method is reliable and
efficient.
PACS: 44.40.+a, 47.70.−n, 52.25.−b
DOI: 10.1088/0256-307X/30/11/114401
In hypersonic re-entry applications, a significant
portion of heating experienced by vehicles can be
caused by radiation due to high speed.[1−3] Since vehicles pass through the atmosphere at high altitude
mostly, the low air density can lead to significant high
temperature and thermal-chemical nonequilibrium.[4]
In these high-temperature nonequilibrium air flows,
the diatomic species may be highly dissociated in the
shock layer, and emission from the two atomic species
N and O is dominant. Therefore, it is highly important to calculate atomic radiation using an efficient
method.
In order to simulate the radiative transfer, radiative properties of nonequilibrium plasmas must be
computed firstly and they play a significant role. The
challenge is to determine the energy level populations
in the nonequilibrium state, which are the basic physical parameters of obtaining radiative properties. Only
the populations of particles are solved to accurately
predict the radiative properties.
Great progress has been made in simulating the
radiative properties of nonequilibrium air plasmas
and the radiative transfer since the 1980s, and there
have been many attempts to couple radiative computation into flow-field codes. Taking thermal and
chemical effects into account and using the correction factor, Carlson[5] presented an approximate
method to calculate nonequilibrium radiation. Nicolet’s RAD/EQUIL code[6] includes models of atomic
and molecular radiation, but the assumption of the
calculation is thermodynamic equilibrium. The widely
used line-by-line (LBL) nonequilibrium air radiation
(NEQAIR) code developed by Park was used to investigate radiative properties of rarefied and nonequilibrium air plasmas by assuming the quasi-steady state
(QSS).[7−9] A structured package for radiation analysis SPRADIAN[10] computes the bound-bound radia-
tion by the LBL method while the others by using corresponding cross sections evaluated by the quantum
defect method. More recently, the collision-radiative
(CR) model is also used to calculate nonequilibrium
air radiation,[11−13] and the improved scheme for the
model has been proposed.[14]
It is known that calculating the radiative properties is a large fraction of the total computational
cost.[15] Considering the CR model as an example,
due to a very large number of processes involved in
the master equation, a full solution is possible but
costly and time-consuming especially in flow-field simulations. Therefore, it is essential in practical applications to develop a fast and efficient method as a
module which solves the radiative properties.
In this Letter, we investigate the nonequilibrium
hot plasmas containing species N and O where the
electronic temperature is not so high (below 20000 K),
and these situations are widely met in various earth
re-entry applications. Our attention focuses on calculating energy level populations. According to the
QSS assumption, an approximate method is presented
on the base of the main idea of the CR model. The
method considers the significant radiative processes
while the modified Saha–Boltzmann distribution is
used as an approximation of the effect of collisional
processes. This approach eliminates the collisional
terms in the master equation and reduces the radiative terms. Thus the solution is timesaving. Then, the
validity of the method is checked by using a two-cell
problem chosen from actual flow-field conditions and
the Stardust flow field, respectively. Finally, computational efficiency is studied by applying the method
to Project Fire II conditions.
According to the QSS assumption, the conservation equation for each electronic level population of
atoms is derived by considering a number of terms
* Supported by the National Natural Science Foundation of China under Grant No 61007047, and the Project of Development
Program for National Defense Technology under Grant No 2011BAK03B07-3.
** Corresponding author. Email: [email protected]
© 2013 Chinese Physical Society and IOP Publishing Ltd
114401-1
CHIN. PHYS. LETT. Vol. 30, No. 11 (2013) 114401
caused by collisional or radiative processes.[16] It is
based on the main idea of the CR model as follows:
the population of an atomic bound energy state may
be changed by collisional and radiative processes. The
former can be subdivided into the cases in which the
impacting particle is an electron, an ion, or a neutral. Moreover, three-body collisional processes are involved. The latter includes spontaneous transition, radiative recombination of ions, photon excitation, and
so on. The model has been investigated in detail in
Ref. [13].
Firstly, collisional terms are substituted by correction factors accompanied by the Saha–Boltzmann
equation when computing nonequilibrium level populations. It is well known that the Saha–Boltzmann
distribution is valid in equilibrium plasmas as a result
of sufficient collisions between particles, so we can consider the effect of all collisional processes to approach
Saha–Boltzmann distribution in nonequilibrium state.
By introducing a correction factor 𝑏𝑖 for the 𝑖th electronic level to the Saha–Boltzmann equation, we obtain
𝑁𝑖 = 𝑏𝑖 𝑁𝑒 𝑁0+
)︁3/2
(︁ 𝐼 )︁
𝑔𝑖 (︁
ℎ2
𝑖
exp
,
+
𝑘𝑇𝑒
2𝑔0 2𝜋𝑚𝑒 𝑘𝑇𝑒
(1)
where 𝑁𝑖 is population of the 𝑖th electronic level, 𝑁𝑒 is
the number density of free electrons, 𝑁0+ and 𝑔0+ are
the number density and degeneracy of charged particles in the ground state, 𝑔𝑖 and 𝐼𝑖 are degeneracy
and ionization energy of the 𝑖th electronic level, 𝑚𝑒 is
the electron mass, 𝑇𝑒 is the electronic temperature, ℎ
is Planck’s constant, 𝑘 is Boltzmann’s constant. By
means of this approach, the collisional terms are eliminated but their effect is represented by the modified
equation.
It is reasonable because, for any arbitrary energy
level population, there always exists a 𝑏𝑖 for the 𝑖th
electronic level to make Eq. (1) correct. If equilibrium
state is studied, we make 𝑏𝑖 equal to 1.0 directly and
Eq. (1) is the Saha–Boltzmann equation exactly. However, departure from 1.0 exists for the correction factor
in nonequilibrium state. Provided that the correction
factor for each electronic level is solved, the energy
level populations are determined by Eq. (1).
Then, radiative terms considered the three main
radiative processes: radiative recombination, spontaneous transition and photo excitation from the ground
state. This can be expressed as
−𝑁𝑖
𝑖−1
∑︁
𝑗=1
∞
∑︁
𝑁𝑗 𝐴𝑗𝑖 +𝑁𝑒 𝑁0+ 𝑅𝑐𝑖 +𝑁𝑔 𝐵𝑔𝑖 𝐼(𝜈𝑔𝑖 ),
(2)
𝑗=𝑖+1
𝐴𝑖𝑗 +
where 𝐴𝑖𝑗 and 𝐴𝑗𝑖 are transition probabilities of
atoms, 𝑁𝑗 is population of the 𝑗 level, 𝑅𝑐𝑖 is the radiative recombination coefficient,[13] 𝑁𝑔 is population of
atoms in the ground state, 𝐵𝑔𝑖 is the Einstein absorption coefficient for the ground state to the 𝑖th level,
𝐼(𝜈𝑔𝑖 ) is the intensity of wavenumber 𝜈𝑔𝑖 correspond-
ing to the transition from the 𝑖th level to the ground
state.
Stimulated absorption from the higher energy levels is neglected because of the estimation as follows.
Setting the electronic temperature 𝑇𝑒 = 20000 K, for a
given higher electronic level 𝑗 (𝑖 > 𝑗 > 1), 𝑁𝑗 𝐵𝑗𝑖 𝐼(𝜈𝑗𝑖 )
is the stimulated absorption from the 𝑗th level to the
𝑖th level. Supposing that Boltzmann distribution is
taken, the proportion [𝑁𝑗 𝐵𝑗𝑖 𝐼(𝜈𝑗𝑖 )]/[𝑁𝑔 𝐵𝑔𝑖 𝐼(𝜈𝑔𝑖 )] is
on the order of 10−2 . For example, Park’s electronic
level system and the ground, the 6th, the 12th level are
investigated,[2] so 𝑖 = 12, 𝑗 = 6. Then Eq. (4) equals
to 1.83%. These estimations are still reasonable for
nonequilibrium state studied here because the departure from equilibrium is not so large. Therefore, neglect of stimulated absorption from higher energy levels may just lead to an error of several percent. Since
the present work focuses on nonequilibrium plasmas
at 𝑇𝑒 ≤ 20000 K, this is applicable for practical calculation.
In addition, we assume that photo absorption from
the ground state to the 𝑖th level equals to spontaneous
transitions from the 𝑖th level to the ground state approximately. Knowing that transitions from high energy level to the ground level emit radiation below
200 nm, this means that the plasma behind the shock
is optically thick for UV radiation below 200 nm.[9] If
the 22 electronic levels for N and 19 levels for O are
applied, as carried out by Park,[2] we will have
𝑁𝑔 𝐵𝑔𝑖 𝐼(𝜈𝑔𝑖 ) ≈ 𝑁𝑖
3
∑︁
𝐴𝑖𝑗 .
(3)
𝑗=1
Finally, from the above analysis we obtain
𝑁𝑖
𝑖−1
∑︁
𝐴𝑖𝑗 =
𝑗=4
∞
∑︁
𝑁𝑗 𝐴𝑗𝑖 + 𝑁𝑒 𝑁0+ 𝑅𝑐𝑖 .
(4)
𝑗=𝑖+1
Equation (4) seems to be not conservative, but it is the
result of introducing correction factors and eliminating collisional terms. Otherwise the collisional terms
might be involved in Eq. (4).
Substituting 𝑁𝑖 and 𝑁𝑗 in Eq. (4) and using
Eq. (1), we have
∑︀∞
2
𝑔
𝐼
𝑅𝑐𝑖 + 𝑗=𝑖+1 𝑏𝑗 𝐴𝑗𝑖 2𝑔𝑗+ ( 2𝜋𝑚ℎ𝑒 𝑘𝑇𝑒 )3/2 exp( 𝑘𝑇𝑗𝑒 )
0
𝑏𝑖 =
.
𝑖−1
∑︀
𝑔𝑖
ℎ2
3/2 exp( 𝐼𝑖 )
(5)
𝐴
+ ( 2𝜋𝑚 𝑘𝑇 )
𝑖𝑗
𝑘𝑇𝑒
𝑒
𝑒
2𝑔
0
𝑗=4
Thus the correction factor 𝑏𝑖 for any state 𝑖 can be
computed by Eq. (5). Here the factor for the highest
level is calculated first, then next to the highest level,
and so on.
To validate the present method, a two-cell problem chosen from actual flow-field conditions was considered first.[17,18] In order to make a comparison, the
parameters we adopted here are the same as Ref. [17].
And Park’s electronic level system is adopted. In this
application, only bound-bound transitions of atoms
are investigated.
114401-2
CHIN. PHYS. LETT. Vol. 30, No. 11 (2013) 114401
there will be no agreements if the population of atoms
are calculated incorrectly.
Next, calculations are carried out in the Stardust
re-entry flight.[21,22] The simulations are made at the
61.8 km altitude in the trajectory, since the temperature is very high and the two atomic species are the
dominant components. The flow-field simulations are
based on Park’s two-temperature model.[2]
K)
In Table 1, a series of correction factors for cells
1 and 2 are calculated by the present method. For
the bottom energy levels, the Boltzmann distribution characterized by the electronic temperature is
assumed,[7−9,19] while the Saha–Boltzmann equilibrium characterized by the electronic temperature is
used for higher energy levels close to free state.[2,20]
The nonequilibrium population is shown by the departure of the correction factor of level 𝑖 from 1.0. For
the seven levels which influence the radiation above
200 nm highly, the results show explicitly that the populations of electronic levels are nonequilibrium.
O
Cell 2
Cell 1
Cell 2
the Boltzmann distribution
3.22
0.24
2.58
0.18
5.74
0.55
5.98
0.60
4.17
0.68
4.24
0.65
0.67
0.12
0.42
0.08
3.35
0.61
66.90
14.55
5.74
1.30
3.89
1.76
24.01
7.38
3.77
1.56
the Saha–Boltzmann distribution (𝑏𝑖 = 1.0)
Cell 1
N
Distance from the body (cm)
Fig. 2. Stagnation line translational and electronic temperature distribution.
2
Table 2. Heat flux exiting the cold cell (W/cm ).
Present
209.42
14.83
224.25
Error (%)
3.35
9.96
4.82
14
12
10
8
6
4
2
0
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2
N
+
N
2
Correlated-𝑘
216.68
16.47
235.61
Radiative heat flux (W/cm )
Number density (10
16
cm
-3
)
Species
N
O
N+O
0.0
O
+
O
60
45
30
15
0
-15
-30
-45
-60
-75
Present
CorrelatedError
-1.0 -0.8 -0.6 -0.4 -0.2
Distance from the body (cm)
10
9
8
7
6
5
4
3
2
1
0.00
Error (%)
𝑖≤5
𝑏6
𝑏7
𝑏8
𝑏9
𝑏10
𝑏11
𝑏12
𝑖 ≥ 13
Temperature (10
level
3
Table 1. Correction factors for electronic levels.
24
20
16
12
8
4
0
-1.2
-1.0 -0.8 -0.6 -0.4 -0.2
Fig. 3. Comparison of heat flux along the stagnation line.
0.0
Distance from the body (cm)
Fig. 1. Stagnation line species number density distribution.
In Table 2, the heat fluxes coming out of the cold
cell results for N, O, or a mixture of them are presented. The errors between correlated-𝑘 method and
the present method are shown at the same time. Ra2
diative heat caused by N comes up to 200 W/cm and
2
is significantly larger than about 15 W/cm caused by
O. Deviation of heat flux due to N is less than 4%
compared with the correlated-𝑘 method, while it is
close to 10% for O. Good accuracy is obtained for
heat flux of the two atomic species with the error less
than 5%. The radiative processes which have been
neglected may cause the deviations. This application
indicates that the present method is reliable, because
Figure 1 shows the number densities of the atomic
species at peak heating along the stagnation streamline. From the presentation, the nitrogen atoms have
the highest number density in the shock layer and
recombine at the capsule surface. The number of
electrons increases rapidly downstream from the bow
shock and then diminishes quickly toward the capsule
surface. The behavior of the charged atoms follows
the electrons similarly.
Figure 2 shows the translational and electronic
temperature distribution along the stagnation line. It
is clear that most of the kinetic energy of the oncoming stream transfers into thermal energy and generates
an aftershock maximum translational temperature up
to 24000 K. Shortly downstream of the bow shock,
the electronic temperature reaches 11000 K, which is
equal to the translational temperature at a distance
of 0.90 cm from the capsule surface approximately.
The discrepancies between translational and electronic
114401-3
CHIN. PHYS. LETT. Vol. 30, No. 11 (2013) 114401
temperature are small within the range of 0.9 cm from
the stagnation point.
Based on the species number density distribution
and temperature distribution obtained (Figs. 1 and 2),
the energy level populations for atomic nitrogen and
oxygen are calculated using the approximate collisionradiative method. The line-by-line method presented
in Ref. [10] is used to solve the radiative transfer equation (RTE). The results of the radiative heat flux along
the Stardust stagnation streamline flow field calculated by both the correlated-𝑘 method and the present
method are shown in Fig. 3. Both the deviations are
plotted. It is quite evident that the results of this
work agree well with the correlated-𝑘 method, and
the maximum error is 7% close to the surface of the
capsule. In a wide range of the shock layer, the error
is smaller than 4%. At the points shortly behind the
bow shock and close to the surface of the capsule, the
error is comparatively larger than others because of
the underestimation of radiative absorption.[23]
In order to access the computational efficiency of
the present method, calculation of the radiative properties and radiative transfer for 50 points along the
stagnation line under the 1634-s conditions of Project
Fire II is carried out, with an uncoupled approach
that the flowfield is solved first to compute the radiative properties and radiative transfer. The CPU
time required on a Lenovo ThinkPad E430 is tested
by means of the present method and the NEQAIR
code, respectively. The present method requires about
19 s to perform this calculation while NEQAIR requires 35 s. Almost a factor of 2 has been obtained
in the improvement of computational efficiency with
the present method. The improvement is achieved by
reducing the coupled terms in the conservation equation of energy levels.
High temperature and low density can make the
diatomic species of air in the shock layer highly dissociated during hypersonic re-entry. For this reason,
it is important to calculate the atomic radiation for
predicting the radiative heat flux. For nonequilibrium
plasmas at relatively low electronic temperature containing species N and O, it is essential to compute the
energy level populations efficiently and accurately for
practical hypersonic re-entry applications in order to
save a lot of computational cost and time.
Based on the investigation on the properties of
air plasmas with the electronic temperature below
20000 K, we simplified the master equation of the full
CR model. By introducing a correction factor into
the Saha–Boltzmann equation, a set of simple conservation equations are obtained. Since the CR model
as the theoretical foundation was tested to be accurate in nonequilibrium and equilibrium, the simplified
presentation is valid for the both with the electronic
temperature below 20000 K.
Furthermore, in order to improve the computational efficiency, the Boltzmann equation using the
electronic temperature is assumed for the bottom lev-
els, while the Saha–Boltzmann equation using the
electronic temperature is assumed for the higher levels. This empirical procedure is supported by the results in the literature mentioned above. With comparatively small errors, these simplifications and assumptions lead to less coupled terms and faster solution.
By means of this simplified method, the correction factors for the energy level populations of atoms
departure from 1.0 to a great extent when studying
highly nonequilibrium plasmas, and they approach 1.0
when the state of the plasmas are close to equilibrium. Due to the simplification, the fluctuation of
the correction factors becomes larger than the fact,
and the correction factors are not equal to 1.0 accurately but approximately 1.0 to a certain extent even
if the nonequilibrium plasmas are investigated. These
fluctuating characteristics have been shown in other
simplified models too.
The present method is validated in extreme cases
by using a two-cell problem chosen from actual
flow-field conditions and the Stardust flow field.
Good agreements are shown in comparison with the
correlated-𝑘 method when simulating the radiative
transfer, which demonstrates that the present approximate method is reliable. Furthermore, as much as 2
times improvement of the computation efficiency compared with NEQAIR shows that the present method
is fast and efficient.
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