Radiation Transfer Modeling in 

Electrothermal Chemical (ETC) Ignition 

by Klaus Kappen 

ARL-CR-506 September 2002

prepared by

Klaus Kappen 

Esterwagnerstraβe 19 A 

85635 Höhenkirchen 

GERMANY 

under contract 

DAAH04-96-C-0086, TCN 01033 

Approved for public release; distribution is unlimited. 
NOTICES 

Disclaimers 

The findings in this report are not to be construed as an official Department of the Army position unless 

so designated by other authorized documents. 

Citation of manufacturer’s or trade names does not constitute an official endorsement or approval of the 

use thereof. 

Destroy this report when it is no longer needed. Do not return it to the originator. 
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-CR-506 September 2002

Radiation Transfer Modeling in 

Electrothermal Chemical (ETC) Ignition 

Klaus Kappen 

Guest Researcher 

prepared by

Klaus Kappen 

Esterwagnerstraβe 19 A 

85635 Höhenkirchen 

GERMANY 

under contract 

DAAH04-96-C-0086, TCN 01033 

Approved for public release; distribution is unlimited. 
Abstract 

The use of plasma ignition for high-performance guns has been pursued at several levels. As 

part of the program at the U.S. Army Research Laboratory (ARL), we are attempting to define

the role of the broadband electromagnetic radiation that is a key energy component of the plasma 

discharge. In particular, this effort is developing a radiation transfer model that will be 

incorporated into the computational fluid dynamics (CFD) model of plasma ignition. This effort 

will allow more physically realistic characterization of the energy transfer in the plasma and its

interaction with the propellant in the models. 

This report summarizes the physical background of the radiation transport, and the validation of 

the different submodules by use of published data will be presented. Current efforts include the 

coupling of the radiation model to the ARL NSRG2 code for the plasma-propellant interaction 

simulation. Furthermore, a series of laboratory experiments will be performed to validate the 

CFD code with the incorporated radiation transport submodel. 

ii
Acknowledgments 

The author would like to thank the U.S. Army Research Laboratory (ARL) for funding this

Scientific Service Agreement under Contract No. DAAH04-96-C-0086, TCN 01033. Thanks

also go to David Mann of the Army Research Office for supporting this contract and to 

Jack Franklin of Battelle, by whom the contract is transacted. The excellent support and 

teamwork of Richard Beyer and Mike Nusca of ARL are highly appreciated. Special thanks go 

to Rose Pesce-Rodriguez, who has done a tremendous job establishing this contract. 

iii
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iv
Contents 

Acknowledgments iii

List of Figures vii

List of Tables vii

1. Introduction 1

2. Physical Background 1

2.1 Equation of State (EOS)..................................................................................................2 

2.2 Absorption Coefficients ..................................................................................................3 

2.3 Method of Partial Characteristics....................................................................................6 

2.4 Three-Dimensional (3-D) Radiation Transfer.................................................................7 

3. Validation 7

3.1 Atomic Data ....................................................................................................................7 

3.2 Net Emission Coefficients.............................................................................................10 

4. Radiation Transport in CFD Simulations 11

5. Summary 12

6. References 13

Report Documentation Page 17

v
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vi
List of Figures 

Figure 1. Calculated LTE number densities of a polyethylene (C2H4) plasma at (a) 1 MPa 

and (b) 10 MPa...........................................................................................................................4 

Figure 2. Comparison of calculated total effective continuum cross section for NI at 

15,000 K. The results of Armstrong et al. [30] have been published by Wilson and 

Nicolet [25]. ...............................................................................................................................9 

Figure 3. Emissivities of hydrogen (a) Ne = 2.9 × 1016 cm!3

 and T = 10,850K and (b) Ne = 

9.3 × 1016 cm!3

 and T = 13,140 K, compared to experimental data of Wiese et al. [10]. 

The thin lines are the contribution due to lines and the continuum. The smooth 

dissolution of the line into the continuum agrees very well with the measurements. .............10 

Figure 4. Net emission coefficients of (a) an isothermal air plasma at different radii and 

(b) an isothermal SF6 plasma of radius 0.1 cm, compared to results of Sevastyanenko 

and Soloukhin [32], Gleizes et al. [33], Liebermann and Lowke [34], and Aubrecht and 

Gross [35]. All calculations have been performed for atmospheric pressure.........................11 

List of Tables 

Table 1. Comparison of some calculated oscillator strengths of neutral carbon CI to data of 

Wiese et al. [29]. .........................................................................................................................8 

Table 2. Comparison of some calculated Stark broadened widths (FWHM) for single ionized 

carbon CII. .................................................................................................................................9 

vii
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viii
1. Introduction 

The plasma ignition of high-loading densities has been investigated for many years and in 

various gun calibers. The advantages of these adjustable igniters (e.g., a temperature 

compensation effect) have been demonstrated in many firings, and the interaction of the plasma

with the propellant has been the focus of many open-air and closed-vessel experiments. Yet, not 

all of the physical and chemical effects of the plasma-propellant interaction (PPI) are understood 

in adequate detail. 

Many experimental investigations have indicated that the energy transfer by plasma radiation is 

an important effect for a fast ignition of the propellant as well as an enhanced mass conversion 

[1–4, 5]. Whether or not it dominates all other energy transport mechanisms from the plasma

source to the propellant will depend strongly on the ignition concept (e.g., Plasma-Jet, Current

Injection) and its parameters (like geometry, pulse duration, pressure levels, etc.). Independent 

of the PPI effects, radiation is the dominant effect of energy transport within the hot plasma. It 

has been known for many years that the ohmic heating of the plasma (by the electrical energy

release) is mainly balanced by radiation. Therefore, radiation transport has to be taken into 

account when solving the energy balance equation in any electrothermal chemical (ETC) 

computational fluid dynamics (CFD) calculations and when studying and simulating PPI effects. 

In this report, the status of a new radiation transport model will be described, which will be

coupled to the ARL CFD code NSRG2 for a more physically realistic description of plasma in it. 

The coupled codes then will be used to study the PPI and, finally, to improve the overall interior

ballistic simulation of ETC guns. In the hot plasma, the radiation by atoms and ions (the 

dissociation and ionization products of the capillary material or eventually of the propellant) 

dominates over molecules, and up to now we have focused on this hot radiation source. But in 

the next step, the absorption by the most important molecules (in the cold boundary layer of the

plasma and the surrounding cold propellant gases) will be implemented as well. An adequate 

description of the radiation transport code, with all of its physical background, would go far 

beyond the scope of this report; only a summary of the basic theories with a focus on some

important aspects is presented next. 

2. Physical Background 

In this section, the physics of plasma radiation is summarized; the reader is referred to some

earlier work [5, 6], the final report of this Short-Term Analysis Service (STAS) contract [7], and 

the references therein for further details. The computation of the radiation transport equation

(RTE) depends on frequency-dependent absorption coefficients κν and emission coefficients εν, 

1
which describe the interaction of the radiation field with matter (i.e., the atoms, ions, and 

molecules). The relation between absorption and emission are given under the assumption of a 

local thermodynamic equilibrium (LTE) by Kirchhoff’s law: 

, (1) 0

ν κν ν ε = I

where is the intensity of a black body radiator at frequency ν and temperature T. Therefore, it 

is sufficient to calculate either the emission or, more commonly, the absorption coefficients, if

LTE is assumed. These absorption coefficients depend on cross sections, σijk, for line and 

continuum absorption and on the number densities, Nijk, of species k in ionization stage j and 

excitation level i. Some aspects in the theoretical calculation of these quantities will be

discussed in the following subsections. 

0

ν I

2.1 Equation of State (EOS) 

An essential difficulty in obtaining equations of state for plasmas arises from the fact that the

Coulomb potential of the ionized species give rise to an infinite number of energy levels. The 

high levels have large radii, which eventually become larger than the mean interparticle distances

in the plasma. Furthermore, the required partition functions then become infinite. For these 

reasons, some cut-off procedures have been introduced in the past. The most common ones are 

based on the Debye or Debye-Hueckel Theory, which leads to a reduction of the ionization

potential. At higher pressures, when the Debye Theory is not valid anymore, often the IonSphere Model is introduced—all levels with a radius larger than the mean interparticle distance 

cannot exist in this theory. Theories which assume a fixed cut off at any exited quantum level 

often lead to useful results when calculating thermodynamic properties of hot gases, but they 

lead to some unphysical behavior when dealing with optical quantities, i.e., the sharp edges of 

photo-ionization cross sections at the threshold of the photo-ionization energy, which never have 

been observed in experimental measurements. Within the framework of the OPACITY Project 

[8], an international collaboration concerned with the calculation of opacities of stellar 

envelopes, Hummer and Mihalas [9] introduced an occupation probability formalism for the 

calculation of the internal partition function: 

Eijk kBT Z w g e

i

ijk ijk

I

jk

− / ≡ ∑ . (2)

Here, gijk is the statistical weight, wijk is the occupation probability, Eijk is the energy of the level,

kB is the Boltzmann constant, and T is the temperature. The wijk go to zero for higher states, 

which ensures that the sum in equation 2 is convergent. Interaction with charged and neutral 

surrounding particles is taken into account. The perturbations by neutral species are based on an 

excluded volume (hard-sphere) treatment (first term on the right of equation 3) and perturbations 

by charged particles are calculated from a fit to a quantum mechanical Stark ionization theory

(using a first-order perturbation theory and a Holtzmark distribution for the ion microfield 

[second term of equation 3]): 

2


































 +  + + 

 



 = − ∑ ∑', '

3/ 2

' ' ' '

3

1/ 2

2

3 0

1

( 1) ( ) 16

3

4 ln

k j

j k j k

ijk ijk

jk

ijk v

v

ijk v N Z

k

Z e

w N r r

χ

π . (3) 

The index ν runs over all neutral species only; r1ν is the ground state radius of the neutral 

perturber, and rijk is the radius associated with state i of species k in ionization stage j of the 

perturbed atom. Zjk is the net charge (Zjk = 0 for neutrals), χijk is the ionization energy, and kijk is

a quantum mechanical correction and the sum in term two runs over all charged particles 

excluding electron. By use of this occupation probability formalism, the number densities in 

each excitation state are then given by a modified Boltzmann equation:

I

jk

ijk B Nijk N jk gijkwijk e Z E k T / − / = . (4)

A comparison of the simulated emission spectrum of the hydrogen Balmer series to experimental

results by Wiese et al. [10] will be shown later. It can be seen that the occupation formalism 

leads to dissolution of the higher lines of the spectral series, resulting in a smooth transition to

the continuum and an excellent agreement with the experimental measurements. 

For the calculation of the ionization equilibrium number densities, an extended version of the 

NASA-Lewis code [11, 12] is used, which has been modified to simulate hot gases and plasmas

[13, 14] (extension to high temperatures, including a collision cross-section database for

calculation of transport coefficients and using a virial-type EOS for high-pressure conditions). The 

code is based on the free-energy minimization procedure in the “chemical picture,” assuming an

LTE. In the future, the particle number densities might be estimated with Nusca’s NSRG2 code

[15], including Anderson’s rate equation submodule [16]. Figure 1 shows calculated number

densities for polyethylene plasmas at temperature and pressure conditions that typically occur in 

current ARL plasma jet experiments [4]. 

2.2 Absorption Coefficients 

The cross section for line absorption (bound-bound) is given by 

( ) (ν ) π

σ ν , ' , '

2

, ' i i i i

e

bb

i i f I

m c

e = ⋅ , (5)

with the charge of an electron e, its mass me, and the velocity of light c; fi,i' is the oscillator 

strength for a transition i → i′ , and Ii,i' is the profile of the line. 

Line profiles for atoms and ions in a plasma are determined by electron impacts and by ion 

microfields. Griem [17] has made extensive calculations for these processes, and Konjević et al.

[18] have reviewed experimental data. It was found that the experimental data are in good 

agreement with calculations made using the methods of Griem. In the semi-empirical electron

3
 (a) (b)

1.0E+19

1.0E+20

1.0E+21

1.0E+22

1.0E+23

1.0E+24

1.0E+25

1.0E+26

1.0E+27

5000 10000 15000 20000 25000 30000

temperature [K]

number density [m^-3]

C I

C III

C IV

H I

H II

H -

C -

H2 +

e - C II

C2

C2H2

C2H

C3

CH2

CH

H2

tot

1.0E+19

1.0E+20

1.0E+21

1.0E+22

1.0E+23

1.0E+24

1.0E+25

1.0E+26

1.0E+27

5000 10000 15000 20000 25000 30000

temperature [K]

number density [m^-3]

C I

C IV

H I

H II

H -

H2 + C -

C III

C2

C2H2

C2H

C3

CH2

CH

H2

to

C II

e -

Figure 1. Calculated LTE number densities of a polyethylene (C2H4) plasma at (a) 1 MPa and (b) 10 MPa. 

impact approximation by Griem [17, 19] and Baranger [20], the full width of the line at half 

maximum (FWHM) is given by 

















< > + < > 













 



 



 Γ = ∑ ∑ ′≠ f ′≠ f

f

i i

i

B

e

e

S i r i g f r f g k T

Ry n

m a

h 2 2

1/ 2

0

3/ 2

| '| | | | '| | | 3 16 π . (6) 

Here, Ry is the Rydberg energy, h is Planck’s constant, is Bohr’s radius, and the matrix 

elements can be expressed as 

a0

, (7) i | i'|r |i | S(J ', J )/ g

2 < > =

with the line strength S for a transition i → i′; gi in equation 7 is the statistical weight of the

ground level, and the sums running over all levels with an allowed transition to the state i ( f ). 

The g values in equation 6 are the so-called effective Gaunt factors [17] for the initial and final 

states 

















∆ ≡

i

B

i i E

k T

g g 2

3 . (8)

The profile of the line in the case of electron impact broadening has a Lorentzian shape 

4
( )2 2

0 ( / 2)

1 ( / 2) ( )

L

L L I − + Γ

Γ = π ω ω

ω , (9)

and when Doppler broadening is also taken into account, the convolution of the two profiles is 

assumed to have a Voigt shape. For further details of the theory of line broadening, the reader is 

referred to the work of Griem and other standard literature. 

For the calculation of line absorption coefficients, the oscillator strengths for all transitions have

to be known. The oscillator strength for a transition i → i′ is 

S

g

E E f

i

i i

i i

( )

3

1 '

'

− → = . (10)

In all cases considered here, a single outer electron participates in the transition. Assuming an 

LS coupling scheme, the line strength may be written as [21] 

. (11) 2 ( ) 1 1 [ , , , ] n l

nl

c R

J L

L S J

L l

L l L

S l J J L L ′ ′

>













′ ′ 











′ ′ = ′ ′

Here the first bracket [.] is shorthand for 

[ ] J , J ′,L,L′ = (2J +1)(2J ′ +1)(2L +1)(2L′ +1) , 

and the {.} brackets refer to Wigner 6-j symbols [21], whose values can be evaluated 

analytically. The designations l ( l' ) refer to the orbital angular momentum of the outer electron, 

with l> being the greater of these; L ( L' ) denotes the total orbital angular momentum, Lc being 

that of the core electrons; and J ( J' ) denotes the total angular momentum of the state i ( i' ).

Finally, the square of the radial integral ( is calculated in the Coulomb approximation of 

Bates and Damgaard [22]. 

) n l Rnl

′ ′

For isolated lines in non-hydrogenic systems, the broadening by ions is of secondary importance. 

Therefore, broadening by ions is not taken into account here, with the following exception: In 

the case of neutral atoms of carbon, nitrogen, and oxygen, Seaton [23] presented a fit formula

that yields results within a factor of 2 of Griem’s [19] published results. This formula is based 

on the previously described line strengths and is used for CI, NI, and OI. Stehlé [24] has 

published tables for Stark broadened hydrogen lines. Her calculation is based on the Model 

Microfield Method (MMM) for both the electronic and ionic broadenings and gives an accurate 

description of the line profile from the center to the wings. These data for the hydrogen Lyman, 

Balmer, and Paschen series are used up to electron number densities of 1019 cm!3

. 

The photoionization cross sections are calculated in the classical theory by Kramers. This theory 

was developed for hydrogen and hydrogen-like species, which means that for the low-ionized 

plasmas in ETC applications the corresponding equations are only correct for hydrogen itself. 

Better agreement for multi-electron systems is obtained, especially for high exited states (which

can be assumed to be hydrogen-like), by use of effective quantum numbers or the effective

5
nuclear charges, respectively. For the photoionization cross sections from the ground states of 

carbon, nitrogen, and oxygen published data of more detailed calculations have been adopted. In 

addition, so called Biberman or Biberman-Norman factors, which describe the difference of 

cross sections for multi-electron species to the hydrogen-like one, have been used for carbon, 

nitrogen, oxygen, and their first ionization stages. As will be shown in section 3, calculated 

cross sections are in good agreement to the published data, e.g., to NI cross sections given by 

Wilson and Nicolet [25]. 

2.3 Method of Partial Characteristics 

The basis for quantitative studies of radiative energy transport is the radiation transport equation 

(RTE). Exact solution of this equation takes many hours of CPU time because thousands of 

absorption coefficients for the continuum and the spectral lines have to be considered for every 

iteration step when solving the energy balance equation. Simple tabulation of the absorption 

coefficients in the presence of spectral lines is also virtually impossible. Therefore, approximate 

but effective methods of integration of radiation characteristics are necessary. One integral 

method that gives values very close to the exact solution is the Method of Partial Characteristics 

(MPC) presented by Sevastyanenko [26, 27]. In this method, the complete description of the 

radiation field is possible by use of precalculated partial intensities 

( ) ( ) ( ) ( ) . (12) ∫ ∫

∞

















∆ = −

0 0

0 ' ' I ξ, X Iν Tξ kν Tξ ,Pξ exp kν Tη,Pη dη dν

x

Here, ∆I(ξ, X) is the contribution to the intensity at point X at temperature TX and pressure pX, 

emitted at point ξ with its local conditions Tξ and pξ and taken into account absorption along the

path 0X . 

The radiation intensity at point X then easily can be estimated by a simple integration along a ray 

0X : 

( ) ( ) , (13) ∫ = ∆

X

I X I X d

0

ξ, ξ

and the radiation flux at X is obtained by integration over all directions 

( ) ( ) ∫∫ = Ω Ω

4π

S X I X d . (14)

Using the MPC, it is possible to generate tables of radiation quantities for rapid calculations of 

radiation transfer in gas-dynamic simulations [5, 28]. A more detailed summary of the MPC can

be found in an earlier paper [6] or in the original work of Sevastyanenko [26, 27]. 

6
2.4 Three-Dimensional (3-D) Radiation Transfer 

The ARL NSRG2 CFD code [15] solves the two-dimensional (2-D)/axisymmetric and timedependent Navier-Stokes equations by use of a finite-volume method. That means for a given 

grid cell, it balances the transfer of mass, momentum, and energy from and to all neighbor grid 

cells. For the radiation field, we have to know not only the contribution from the neighbor grid 

cells, but also the contribution from all cells having a line-of-sight to the cell of interest. For this

reason, a quasi-3-D integration method has been developed, which integrates the partial radiation 

intensities of the whole 3-D volume for every grid point of the 2-D CFD grid, i.e., all grid points 

ξ'(r', φ, z') of the 3-D volume having a line-of-sight to a grid point X (r, z) of the 2-D CFD grid 

are taken into account. The necessary conditions at any point of the 3-D volume, e.g., the 

temperature, can easily be estimated from the 2-D grid, if the radial and longitudinal coordinates 

r' and z' of the point ξ' are known. A detailed description of this quasi-3-D integration 

submodule will be given elsewhere [7]. This 3-D integration submodule, together with some

routines for the interpolation of radiation characteristics from the databases, are the only parts of 

the overall radiation code which will be integrated in the CFD code. 

3. Validation 

The final ARL plasma flow and PPI simulation code will contain highly sophisticated 

submodules like the radiation transport module, a module for solving the required rate equations, 

an ablation, and ignition module, etc. The validation of the coupled modules is a difficult part, 

for which reliable experimental data have to be available, too. A better way than to validate the

overall program in a single step is to validate the different submodules as stand-alone programs 

first. In the case of the radiation part, the results of some comparisons to published theoretical 

and experimental data will be shown. 

3.1 Atomic Data 

About 150 calculated oscillator strengths and about 100 calculated line widths have been 

compared to the extensive compilation of the U.S. National Institute of Standards and

Technology (NIST) [29] and the standard reference tables of Griem [17] for the atoms and ions

of C, N, and O (for hydrogen the theories are exact). Almost all values are within a factor of 2 to 

the references, whereas most of them deviate only by a maximum of 20%. For example, Tables 1 

and 2 show some results for the oscillator strengths of neutral carbon CI and the Stark 

broadening of single ionized carbon CII. 

7
Table 1. Comparison of some calculated oscillator strengths of neutral carbon CI to data of Wiese et al. [29]. 

Transition

nl → n′l′

Multiplat gi – gf Ei Ef 

(cm!1) (cm!1) 

fi!f 

(this work) 

fi!f 

(Wiese et al. [29])

2p – 3s 3P – 3Po 3 – 5 

1 – 3 

5 – 5 

3 – 3 

3 – 1 

5 – 3 

16.40 – 60393.14

 0.00 – 60352.63

43.40 – 60393.14 

16.40 – 60352.63 

16.40 – 60333.43 

43.40 – 60352.63 

6.615E!02

1.592E!01

1.191E!01

3.981E!02

5.316E!02

3.982E!02

5.88e!02

1.40e!01

1.04e!01

3.56e!02

4.71e!02

3.56e!02

2p – 3s 1S – 1Po 1 – 3 21648.01 – 61981.82 1.251E!01 9.40e!02

3s – 4p 3Po – 3P 1 – 3 

3 – 5 

3 – 3 

3 – 1 

5 – 5 

5 – 3 

60333.43 – 81325.76

60352.63 – 81343.99

60352.63 – 81325.76

60352.63 – 81311.01

60393.14 – 81343.99

60393.14 – 81325.76

5.247E!03

2.294E!03

1.298E!03

1.648E!03

4.041E!03

1.269E!03

3.44e!03

1.54e!03

8.04e!04

1.22e!03

2.72e!03

9.94e!04

3s – 3p 3Po – 3D 3 – 5 

1 – 3 

5 – 7 

3 – 3 

5 – 5 

5 – 3 

60352.63 – 69710.66

60333.43 – 69689.48

60393.14 – 69744.03

60352.63 – 69689.48

60393.14 – 69710.66

60393.14 – 69689.48

3.757E!01

5.004E!01

4.215E!01

1.250E!01

7.502E!02

4.990E!03

3.99e!01

5.34e!01

4.42e!01

1.29e!01

7.60e!02

4.95e!03

3s – 4p 1Po – 1P 3 – 3 61981.82 – 80562.85 5.006E!03 8.08e!03

3d – 6s 3D – 3Po 3 – 5 

5 – 5 

3 – 3 

3 – 1 

7 – 5 

5 – 3 

69689.48 – 86369.60

69710.66 – 86369.60

69689.48 – 86331.63

69689.48 – 86321.94

69744.03 – 86369.60

69710.66 – 86331.63

1.886E!04

1.700E!03

2.829E!03

3.768E!03

6.820E!03

5.100E!03

1.93e!04

1.74e!03

2.89e!03

3.85e!03

6.93e!03

5.21e!03

Note: Many thousands values for the atoms and ions of hydrogen, carbon, nitrogen, and oxygen are taken

into account in the radiation transport code. 

In Figure 2, a comparison of a calculated continuum spectrum for neutral nitrogen to results by 

Armstrong et al. (published by Wilson and Nicolet [25]) is shown. Good agreement is obtained 

and, whereas the spectrum of Armstrong shows the sharp edges at the photoionization 

thresholds, the present results using the occupation probability formalism are more physical. 

To get good and reliable experimental data is often as difficult as it is to calculate them. Wiese

et al. [10] published experimental data of emissivities of the hydrogen Balmer series, which have

often been used for validation of theory and simulation. In Figure 3, current simulation results 

are compared to these measured data at two different temperatures and electron number

densities. In addition to free-free Bremsstrahlung and bound-free recombination of HI, the photo 

attachment to the negative hydrogen is included in the simulation. Both theoretical and 

experimental emissivities are absolute, i.e., no scaling has been applied. 

8
Table 2. Comparison of some calculated Stark broadened widths (FWHM) for single ionized carbon CII. 

Transition

nl → n′l′

Multiplat λ

(Å) 

T 

104 

(K) 

Ne

1016

(cm!3

) 

γ

a

(Å) 

γ

b

(Å) 

γ

c 

(Å) 

2p – 3s 2Po – 2S 858 10

20 

10.0 

10.0 

6.699e!3 

4.737e!3 

6.889e!3 

4.871e!3 

1.650e!2 

1.300e!2 

3s – 3p 2S – 2Po 6580 10

20 

10.0 

10.0 

8.324e!1 

5.886e!1 

8.371e!1 

5.919e!1 

1.192e+0

1.038e+0

3s – 4p 2S – 2Po 2174 20 10.0 2.130e!1 2.267e!1 2.960e!1 

3p – 3d 2Po – 2D 7234 20 10.0 8.669e!1 1.068e+0 1.198e+0

3p – 4s 2P – 2S 3920 20 10.0 8.604e!1 6.492e!1 9.600e!1 

3p – 4d 2Po – 2D 2747 20 10.0 1.153e+0 1.392e+0 9.360e!1 

3d – 4p 2D – 2Po 5890 20 10.0 1.667e+0 1.896e+0 2.180e+0

3d – 4f 2D – 2Fo 4267 20 10.0 1.259e+0 2.571e+0 1.690e+0

3d – 5p 2D – 2Po 3361 20 10.0 1.999e+0 2.754e+0 1.836e+0

3d – 5f 2D – 2Fo 2993 10

20 

10.0 

10.0 

3.567e+0

3.060e+0

4.919e+0

5.241e+0

4.560e+0

3.840e+0

a

Comparison made by method described in text. 

bComparison made by a simpler approximation for the matrix elements. 

c

Comparison made with values given by Griem [17] (core configuration is 2s2(

1

S)). 

Note: Many thousands widths for the lines of atoms and ions of hydrogen, carbon, nitrogen, and oxygen

are taken into account in the radiation transport code. 

1.0E-21

1.0E-20

1.0E-19

1.0E-18

1.0E-17

1.0E-16

0 4 8 12 16

Photon energy [eV]

effective continuum cross section 

[cm^2]

Kappen

Armstrong

20

Figure 2. Comparison of calculated total effective continuum cross section for NI at 15,000 K. The

results of Armstrong et al. [30] have been published by Wilson and Nicolet [25]. 

9
10

Figure 3. Emissivities of hydrogen (a) Ne = 2.9 × 1016 cm!3 and T = 10,850K and (b) Ne = 9.3 × 1016 cm!3

and T = 13,140 K, compared to experimental data of Wiese et al. [10]. The thin lines are the

contribution due to lines and the continuum. The smooth dissolution of the line into the 

continuum agrees very well with the measurements. 

3.2 Net Emission Coefficients 

To validate the integrated radiation characteristics stored as databases, net emission coefficients 

have been calculated and compared to published results. Lowke [31] has shown that the 

averaged radiation intensity of an isothermal cylinder with radius is approximately the same

as for an isothermal sphere. By using the simpler formula for the sphere, the net emission 

coefficient is given as 

R

∫

∞

= ′ − ′

0

0 ε N Iν kν exp( kνR)dν

15.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

3600 4100 4600 5100

Wavelength [Å]

log [photons cm^-3 s^-1 Å^-1]

Wiese et al.

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19.5

3600 4100 4600 5100

Wavelength [Å]

log [photons cm^-3 s^-1 Å^-1]

Wiese et al.

(a) (b) 

. 

From equations 12 and 13 it can easily be seen that the net emission coefficient can be estimated 

from the radiation characteristics database if we use an isothermal temperature profile with Tx = Tξ

and radius X = R. In Figure 4, a comparison of net emission coefficients for high temperature air 

and SF6

*

 is shown. 

 

*

 The calculations of SF6 have only been performed for validation purposes; there is no interest in this gas for ETC applications.
Figure 4. Net emission coefficients of (a) an isothermal air plasma at different radii and (b) an isothermal 

SF6 plasma of radius 0.1 cm, compared to results of Sevastyanenko and Soloukhin [32], 

Gleizes et al. [33], Liebermann and Lowke [34], and Aubrecht and Gross [35]. All calculations 

have been performed for atmospheric pressure. 

Current efforts are the validation of the 3-D model. First results indicate that it is working well, 

so in the next step of the STAS program an interface between the RT code and the CFD code

will be set up and tested.

4. Radiation Transport in CFD Simulations 

( v ) v 0 ( )+ ∇⋅ + ⋅∇ +∇ +∇ = ∂

∂

e p q S

t

e r r r r ρ ρ

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

5000 10000 15000 20000 25000 30000

temperature [K]

net emission coefficient [W cm^-3 sr^-1]

0.01 cm - Kappen

0.01 cm - Sevast'yanenko

 0.1 cm - Kappen

 0.1 cm - Sevast'yanenko

 1.0 cm - Kappen

 1.0 cm - Sevast'yanenko

1.0E+0

1.0E+1

1.0E+2

1.0E+3

1.0E+4

8000 12000 16000 20000 24000

temperature [K]

net emission coefficient [W cm^-3 sr^-1]

Kappen

Gleizes et al.

Liebermann and Lowke

Aubrecht and Gross

(a) (b) 

There are two aspects of radiation in ETC CFD simulation: first, the energy transfer by radiation 

in the hot plasma and gas, and second, the interaction of the radiation energy that is transmitted

to the capillary wall or propellant. The current ARL CFD codes do not include the ohmic 

heating in the energy balance, i.e., the equation that is solved may be written as [15] 

, (15)

with the den with the density ρ, the velocity v

r , the energy e, the pressure p, and the thermal heat flux q

r

. The

divergence of the radiation flux r will be calculated directly by use of the MPC without 

numerical differentiation. From the inner hot core of the plasma to the outer cold gas zone, this 

S

S

r

∇ term will change its sign: whereas the radiation is mainly responsible for energy losses in

11
the hot core, a lot of radiation energy will be reabsorbed at the cold boundary and the 

surrounding gas, leading to a temperature increase in this regions. Regarding the interface, only

the 3-D integration submodule together with some interpolation routines and the MPC databases 

will be coupled to the CFD code. In every iteration and time step when the energy balance is

solved, the CFD code will call the radiation submodule with the actual pressure and temperature 

profile (for all grid points), and the divergence of the radiation flux will be given back. 

When taking into account material ablation in the capillary and/or ignition of a propellant, in 

addition to the S

r

∇ values the radiation energy flux at the wall (or grains) is required. Calling the 

RT code for a second time, which in that case will use a different integration algorithm [6], this 

can be done. The resulting radiation energy will then be used for PPI studies, in new pyrolysis 

laws and/or ablation modules in the CFD simulation. Regarding propellants, which may be

transparent for parts of the electromagnetic spectrum, one might be interested in the energy 

transfer within different frequency bands. Using the presented radiation transport code, either 

the total energy transfer from the vacuum ultraviolet (VUV) to the infrared (IR) or by radiation

of some frequency bands of special interest can be calculated.

5. Summary 

A new radiation transport code has been described, which will be used in the ARL NSRG2 code

for the plasma-propellant interaction simulation. With the implemented method of partial 

characteristics it is possible to precalculate databases, which allow one to calculate the energy 

transfer by radiation iteratively at the same time when solving the balance equations of mass, 

momentum, and energy in any CFD code in an acceptable CPU time. Some efforts for the 

validation of the program have been presented, and it has been shown that the current version of 

the code yields results, which are in very good agreement to published theoretical as well 

experimental data. In addition, a 3-D integration method has been implemented, which allows

one to use the code for calculations in any cylinder-symmetrical volume (the precalculated 

partial intensities stored in the databases are independent of the geometry); the validation of this 

part is almost finished. In a next step, absorption of the most important molecules will be added 

to the code and current efforts are the coupling of the RT code to the ARL NSRG2 code. The 

focus of the current activities is on plasmas containing molecules, atoms, and ions of hydrogen, 

carbon, nitrogen, and oxygen, but the program is in no way limited to this. Finally, the 

performance of some laboratory experiments is underway to validate the CFD code with 

incorporated radiation simulation. 

Until now, the radiation transport code consists of more than 40 subroutines and 6 input 

databases/files for the energy level, quantum numbers, etc. Finally, a graphical user interface

(GUI) has been developed, which allows one to use the code without extensive knowledge of 

quantum mechanics and plasma physics. 

12
6. References 

1. White, K. J., G. L. Katulka, and S. Driesen. “Electrothermal-Chemical Plasma Interaction

With Propellants.” 32nd JANNAF Combustion Subcommittee Meeting, CPIA Publication

631, 1995.

2. Koleczko A., W. Erhardt, S. Kelzenberg, and N. Eisenreich. “Plasma Ignition and 

Combustion.” 32nd International Annual Conference of ICT, 2001. 

3. Koleczko, A., and N. Eisenreich. Personal communications. Fraunhofer Institute for 

Chemical Technologies, Pfinztal, Germany, 1999. 

4. Beyer, R. A. “Small Scale Experiments in Plasma Propellant Interactions.” Proceedings of 

the 37th JANNAF Combustion Subcommittee Meeting, CPIA Publication 701, vol. 1, p. 137, 

2000. 

5. Kappen, K., and U. H. Bauder. “Calculation of Plasma Radiation Transport for Description

of Propellant Ignition and Simulation of Interior Ballistics in ETC Guns.” IEEE 

Transactions on Magnetics, vol. 1, 2001. 

6. Kappen, K., and U. H. Bauder. “Simulation of Plasma Radiation in ElectrothermalChemical Accelerators.” IEEE Transactions on Magnetics, vol. 35, no. 1, 1999. 

7. Kappen, K. STAS Contract DAAH04-96-C-0086. Unpublished data. 

8. Seaton, M. J. “Opacities for Stellar Envelopes.” Monthly Notices of the Royal Astronomical 

Society, vol. 266, p. 805, 1994. 

9. Hummer, D. G., and D. Mihalas. “An Occupation Probability Formalism for the Truncation

of the Internal Partition Function.” The Astrophysical Journal, vol. 331, p. 794, 1988. 

10. Wiese, W. L., D. E. Kelleher, and D. R. Paquette. “Detailed Study of the Stark Broadening 

of Balmer Lines in High-Density Plasma.” Physical Review A, vol. 6, no. 3, p. 1132, 1972. 

11. Gondon, S., and B. J. McBride. “Computer Program for the Calculation of Complex 

Chemical Equilibrium Compositions, Rocket Performance, Incident and Reflected Shocks, 

and Chapman-Jourguet Detonations.” NASA-SP-273, NASA-Lewis Research Center, 

Cleveland, OH, 1971. 

12. Svehla, R. A., and B. J. McBride. “Fortran IV Computer Program for the Calculation of 

Thermodynamic and Transport Properties of Complex Chemical Systems.” NASA-TN-D-7056, 

NASA-Lewis Research Center, Cleveland, OH, 1973. 

13
13. Kovitya, P. “Theoretical Determination of Material Functions of Plasmas Formed of 

Ablated PTFE, Aluminium, PVC, and Perspex for Temperature Range of 5000 to 30000 K.”

CSIRO Division of Applied Physics TM No. 3, Lindfield, New South Wales, Australia, 

1982. 

14. Kappen, K. “Berechnung der Materialfunktionen von Hochdruckplasmen am Beispiel von 

Methanol.” Diploma thesis, Technical University Munich, 1995. 

15. Nusca, M. J., M. J. McQuaid, and W. R. Anderson. “Development and Validation of a

Multi-Species Reacting Flow Model for the Plasma Jet Generated by an ETC Igniter.” 

Proceedings of the 37th JANNAF Combustion Subcommittee Meeting, CPIA Publication

701, p. 1, 2000. 

16. Nusca, M. J., and W. R. Anderson. “CFD Model of the Plasma Efflux Generated by an ETC 

Igniter: Characterization of Propellant Surface Conditions.” Proceedings of the 38th 

JANNAF Combustion Subcommittee Meeting, 2002. 

17. Griem, H. R. Spectral Line Broadening by Plasmas. New York: Academic Press, 1974. 

18. Konjević, N., M. S. Dimitrijević, and W. L. Wiese. Journal of Physical and Chemical 

Reference Data. Vol. 13, p. 619, 1984. 

19. Griem, H. R. “Semiempirical Formulas for the Electron-Impact Widths and Shifts of 

Isolated Ion Lines in Plasmas.” Physical Review, vol. 165, no. 1, p. 258, 1968. 

20. Baranger, M. Physical Review. Vol. 111, p. 494, 1958. 

21. Cowan, R. D. The Theory of Atomic Structure and Spectra. Berkeley: University of 

California Press, 1981. 

22. Bates, D. R., and A. Damgaard. Philosophical Transactions of the Royal Society of London, 

Series A: Mathematical and Physical Sciences, p. 242, 1949. 

23. Seaton, M. J. “Atomic Data for Opacity Calculations: XII. Line-Profile Parameter for 

Neutral Atoms of He, C, N, and O.” Journal of Physics B: Atomic Molecular and Optical 

Physics, vol. 22, p. 3603, 1989. 

24. Stehlé, C., and R. Hutcheon. “Extensive Tabulations of Stark Broadened Hydrogen Line 

Profiles.” Astronomy and Astrophysics Supplement, vol. 144, p. 93, 1999. 

25. Wilson, K. H., and W. E. Nicolet. “Spectral Absorption Coefficients of Carbon, Nitrogen, 

and Oxygen Atoms.” Journal of Quantitative Spectroscopy & Radiative Transfer, vol. 7, 

p. 891, 1967. 

26. Sevastyanenko, V. G. “Radiation Transfer in a Real Spectrum: Integration Over 

Frequency.” Inzhenerno-fizicheskii Zhurnal, vol. 36, no. 2, 1979. 

14
27. Sevastyanenko, V. G. “Radiation Transfer in a Real Spectrum: Integration With Respect to 

Frequency and Angles.” Inzhenerno-fizicheskii Zhurnal, vol. 38, no. 2, 1980. 

28. Gruber, K., K. Kappen, A. Voronov, and H. Haak. “Radiation Absorption of Propellant 

Gas.” IEEE Transactions on Magnetics, vol. 1, 2001. 

29. Wiese, W. L., J. R. Fuhr, and T. M. Deters. “Atomic Transition Probabilities of Carbon, 

Nitrogen, and Oxygen—A Critical Data Compilation.” Journal of Physical and Chemical 

Reference Data, monograph no. 7, 1996. 

30. Armstrong, B. H., R. R. Johnston, and P. S. Kelly. “Opacity of High Temperature Air.” 

Lockheed Missiles & Space Co. Report No. 8-04-64-2, 1964.

31. Lowke, J. J. “Predictions of Arc Temperature Profiles Using Approximate Emission 

Coefficients for Radiation Losses.” Journal of Quantitative Spectroscopy & Radiative

Transfer, vol. 14, p. 111, 1973. 

32. Sevastyanenko, V. G., and R. I. Soloukhin. Handbook of Radiative Heat Transfer in HighTemperature Gases. New York: Hemisphere Publishing Corporation, 1987. 

33. Gleizes, A., B. Rahmani, J. J. Gonzales, and B. Liani. Journal of Physics D: Applied 

Physics. Vol. 24, p. 1300, 1991. 

34. Liebermann, R. W., and J. J. Lowke. Journal of Quantitative Spectroscopy & Radiative 

Transfer. Vol. 16, p. 253, 1976. 

35. Aubrecht, V., and B. Gross. “Net Emission Coefficient of Radiation in SF6 Arc Plasmas.”

Journal of Physics D: Applied Physics, vol. 27, p. 95, 1994. 

15
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16
17

plasma, propellant, ignition, radiation

The use of plasma ignition for high-performance guns has been pursued at several levels. As part of the program at the

U.S. Army Research Laboratory (ARL), we are attempting to define the role of the broadband electromagnetic radiation

that is a key energy component of the plasma discharge. In particular, this effort is developing a radiation transfer model

that will be incorporated into the computational fluid dynamics (CFD) model of plasma ignition. This effort will allow

more physically realistic characterization of the energy transfer in the plasma and its interaction with the propellant in

the models.

This report summarizes the physical background of the radiation transport, and the validation of the different

submodules by use of published data will be presented. Current efforts include the coupling of the radiation model to

the ARL NSRG2 code for the plasma-propellant interaction simulation. Furthermore, a series of laboratory experiments

will be performed to validate the CFD code with the incorporated radiation transport submodel.

The author is a Guest Researcher at ARL.

ARL-CR-506

Esterwagnerstraße 19 A

85635 Höhenkirchen

GERMANY 

Klaus Kappen

C: DAAH04-96-C-0086,

TCN 01033, D.O. 0699

Radiation Transfer Modeling in Electrothermal Chemical (ETC) Ignition

September 2002 Interim Report, June 2001 – March 2002

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gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this

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