# Thermodynamic and Transport Properties of Two-temperature Oxygen Plasmas

- 241 Downloads
- 31 Citations

## Abstract

Thermodynamic and transport properties of two-temperature oxygen plasmas are presented. Variation of species densities, mass densities, specific heat, enthalpy, viscosity, thermal conductivity, collision frequency and electrical conductivity as a function of temperature, pressure and different degree of temperature non-equilibrium are computed. Reactional, electronic and heavy particle components of the total thermal conductivity are discussed. To meet practical needs of fluid-dynamic simulations, temperatures included in the computation range from 300 K to 45,000 K, the ratio of electron temperature (*T* _{e}) to the heavy particle temperature (*T* _{h}) ranges from 1 to 30 and the pressure ranges from 0.1 to 7 atmospheres. Results obtained for thermodynamic equilibrium (*T* _{e} = *T* _{h}) under atmospheric pressure are compared with published results obtained for similar conditions. Observed overall agreement is reasonable. Slight deviations in some properties may be attributed to the values used for collision integral data and for the two temperature formulations used. An approach for computing properties under chemical non-equilibrium and associated deviations from two-temperature results under similar conditions are discussed.

## Keywords

Non-equilibrium thermodynamic and transport properties oxygen plasmas## Introduction

Oxygen as plasma gas in plasma cutting torches offers higher quality surfaces inside the kerf, better removal of molten metal, less dross formation beneath the work piece and faster cutting speed. To meet the industrial requirements of precision, cost, quality and quantity and to compete with existing technologies, new developments consider smaller nozzle diameters, higher currents and higher flow rate of gases. Enormous constriction of the arc (nozzle diameter ∼2 mm), extremely high temperatures (> 30,000 K) at the core, extremely sharp gradient of temperature (≥ 10^{4} K/mm) and associated high flow fields (*M* > 1) promote in these new generation of plasma torches significant deviations from thermal and chemical equilibrium. While, optimum designs of such plasma devices definitely require a thorough understanding of the impact of device design parameters on the ultimate fluid dynamic and electromagnetic characteristics, availability of non-equilibrium properties of oxygen plasma is a prerequisite. This paper addresses this issue and offers a complete study on the non-equilibrium transport and thermodynamic properties of oxygen plasmas necessary for such investigations.

Transport properties of thermal plasmas have been extensively studied [1, 2, 3, 4, 5, 6, 7, 8] for various pure as well as binary mixtures of plasma gases under local thermal equilibrium (LTE) conditions, based on the solution of the Boltzmann equation using the Chapman-Enskog’s [9, 10] approach. Assuming a complete decoupling between electrons and heavy particles, Devoto [11, 12] and later Miller [13], Kannappan [14] and Bonnefoi [15] have derived transport coefficients of two temperature plasmas. Aubreton et al. [16] have reported two-temperature transport properties of Ar–O_{2} plasmas under atmospheric pressure. Recently, it has been shown by Rat et al. [17, 18] that due to the assumption of complete decoupling between electrons and heavy particles, values of diffusion coefficients used by Devoto in earlier computations were not consistent with mass conservation. Afterward, two-temperature properties including diffusion have been reported by Rat et al. [19, 20, 21] for argon plasmas and by Aubreton et al. [22] for argon-helium plasmas.

While, most of the reported results are for atmospheric pressure and include temperature ranges below 30,000 K and θ(= *T* _{e}/*T* _{h}) values ≤ 10, no two-temperature properties are available for pure oxygen plasmas. Moreover, inside of such plasma torches, pressure can practically vary from several atmospheres to sub-atmospheric and there is increasing experimental evidence that temperature inside constricted nozzles (∼2 mm dia), plasma temperatures at higher currents (∼200 A) may well exceed 30,000 K. Also, near the wall, the non-equilibrium parameter θ can reach values beyond 20. Another reason for considering temperatures beyond 30,000 K is to achieve convergence in numerical simulations. It has been observed that even though the maximum temperature in a final converged solution may not exceed 30,000 K, it may easily exceed this limit at some intermediate stages of the computation. Unavailability of property data, physically compatible with the associated temperature variation, often leads to divergence of the code.

In this paper we have presented a complete set of thermodynamic and transport properties as required for numerical simulation of non-equilibrium oxygen plasmas at higher currents. Necessary collision integral values for computations are either collected from the literature or calculated. Cases, where integral values are available only for lower ranges of temperature, data have been extrapolated based on the available data. To take proper account of multicomponent diffusion and thermal diffusion in multi-temperature plasmas, results from the study of Ramshaw [23, 24] have been used. Under chemical non-equilibrium, number densities inside the plasma are influenced by diffusive particle fluxes originating from species and temperature gradients. The ultimate effect can be accounted for by including a modified rate constant in the associated rate equations in place of chemical-equilibrium rate constants. A chemical non-equilibrium parameter ‘*r*’ is defined as the ratio of the modified rate constant to the equilibrium rate constant. Based on practical needs, ranges of pressure, temperature and θ are selected to vary respectively from 0.1 atm to 7 atm, 300 K to 45,000 K and from 1 to 30.

Details of the computational approach are presented in Section ‘Method used for determination of thermodynamic properties of two temperatur oxygen plasmas’. For θ = *r* = 1 results correspond to chemical as well as thermal equilibrium. Properties are available from previously published reports [3] and can be readily compared for this case. Such comparisons are presented in Section ‘Method used for determination of transport properties of two temperature oxygen plasmas’ together with other major results obtained from the study. Conclusions are presented in Section ‘Results’.

## Method Used for Determination of Thermodynamic Properties of two Temperature Oxygen Plasmas

- (1)
O

_{2}= O + O - (2)
O

^{+}= O−e - (3)
O

^{++}= O^{+}−e

*T*

_{e}) while that of the rotational-vibrational levels of molecules are governed by the heavy particle temperatures (

*T*

_{h}). In the system all the electrons follow a temperature

*T*

_{e}and all other particles except electrons follow a temperature

*T*

_{h}. The total partition function of a species ‘i’ is the product of its internal (

*Q*

_{int}

^{i}) and translational partition functions (

*Q*

_{tr}

^{i}). The internal partition function is a product of rotational (

*Q*

_{rot}

^{i}), vibrational(

*Q*

_{vib}

^{i}), electronic (

*Q*

_{el}

^{i}) and nuclear partition functions. Contributions from nuclear partition functions are neglected in the formulation since the nuclei play no part in associated chemical processes. Under this assumption, the internal and translational partition functions of involved species are listed in Table 1.

*Q*

_{int}

^{i},

*Q*

_{tr}

^{i},

*Q*

_{rot}

^{i}, and

*Q*

_{vib}

^{i}respectively present internal, translational, rotational and vibrational partition function of species i. m

_{i}, k

_{B}and h respectively correspond to mass of species i, Boltzmann constant and Planck’s constant.

Internal and translational partition functions for the used model

Species | Internal partition function | Translational partition function [V = volume of system] |
---|---|---|

M | | \({Q_{\rm tr}^{\rm M}=V \cdot \left({\frac{2\pi m_{\rm M} k_{\rm B} T_{\rm h}}{h^{2}}}\right)^{3/2}}\) |

A | | \({Q_{\rm tr}^{\rm A}=V \cdot \left({\frac{2\pi m_{\rm A} k_{\rm B} T_{\rm h}}{h^{2}}}\right)^{3/2}}\) |

I | | \({Q_{\rm tr}^{\rm I}=V \cdot \left({\frac{2\pi m_{\rm I} k_{\rm B} T_{\rm h}}{h^{2}}}\right)^{3/2}}\) |

D | | \({Q_{\rm tr}^{\rm D}=V \cdot \left({\frac{2\pi m_{\rm D} k_{\rm B} T_{\rm h}}{h^{2}}}\right)^{3/2}}\) |

e | | \({Q_{\rm tr}^{\rm e}=V \cdot \left({\frac{2\pi m_{\rm e} k_{\rm B} T_{\rm e}}{h^{2}}}\right)^{3/2}}\) |

_{rot}and Θ

_{vib}are respectively characteristic rotational temperature (=2.09 K for Oxygen) and vibrational temperature (=2,260 K for Oxygen).

*E*

_{ij}refers to the energy of jth electronic excited level of species i and g

_{ij}corresponds to its degeneracy.

Appropriate values for *E* _{M}, *E* _{I} and *E* _{D} for oxygen plasma are respectively: 5.115 eV, 13.614 eV and 48.722 eV. *E* _{I}−*E* _{A} presents the energy required for first ionization and so on. *p* is the total pressure in which *p* _{i} is the partial pressure of species i. Lowering of the ionization energy [28] due to the interaction of charged particles has been included in the final computation.

## Method Used for Determination of Transport Properties of Two Temperature Oxygen Plasmas

For calculating transport properties, the distribution functions followed 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 [9], which finally through linearization of the Boltzmann equation leads to a system of linear equations that can be solved to obtain different transport coefficients [10, 29]. While the elements in the matrix of the system of equations depend on the interaction between associated colliding species in terms of collision integrals, the number of elements depends on the order of the chosen approximation. In the following we describe in brief the definition of collision integrals and prescription used for computing different transport properties from them.

### Collision Integrals

*g*is the relative speed between colliding particles,

*Q*

_{ij}

^{(l)}(

*g*) is the gas kinetic cross-section which is a function of impact parameter, deflection angle and associated interaction potential [9]. γ

^{2}= m

^{*}

_{ij}

*g*

^{2}/2

*k*

_{B}

*T*

_{ij}

^{*}and m

_{ij}

^{*}and

*T*

_{ij}

^{*}are the reduced mass and temperature i.e.:

_{ij}

^{(1,1)}, Ω

_{ij}

^{(1,2)}, Ω

_{ij}

^{(1,3)}and Ω

_{ij}

^{(2,2)}. Collision integrals associated with O

^{+}−O

_{2}, O

^{++}−O

_{2}, and O

^{++}−O interactions are computed using polarization potentials following suggestions of Devoto [1]. The remaining integral values are available [31, 32, 33, 34] in the literature in form of reduced integrals as listed in Table 2. However, covered ranges of temperature are somewhat limited in these references except for [31]. To cover the range of our interest (300K-45000K), available integral values are extrapolated based on the available data. For charged species, collision integrals are computed using screened Coulomb potentials according to Liboff [35]. Recursion formulae [32] have been used for derivation of Ω

_{ij}

^{(l,s+1)}integrals from Ω

_{ij}

^{(l,s)}integrals for O

_{2}−O

_{2}and O

_{2}−O interactions. For electron neutral interactions, following Murphy [3], it has been assumed that Ω

_{ij}

^{(l,s)}= Ω

_{ij}

^{(1,1)}.

Sources of collision integral data/formulation

Interaction | Method | Reference |
---|---|---|

O–O | Collision integral tabulation (CIT) | Levin [31] |

O | CIT | Yun [32] |

O | CIT | Yun [32] |

O | Polarization potential | Devoto [1] |

O | Polarization potential | Devoto [1] |

O | CIT | Stallcop [33] |

O | Polarization potential | Devoto [1] |

e–O | CIT | Devoto [34] |

e–O | CIT | Devoto [34] |

### Diffusion Coefficients

Here, *T* _{i} and *T* _{j} are temperatures of ith and jth particle respectively. *p* is the pressure, Ω _{ij} ^{(1,1)} is the collision integral and *k* _{B} is the Boltzmann constant. Reduced mass *m* _{ij} ^{*} and reduced temperature *T* _{ij} ^{*} are used as defined in (10). Once the temperatures are known, twenty five binary diffusion coefficients associated with this case are computed in this manner using already available Ω _{ij} ^{(1,1)} values. Subsequently, all other diffusion coefficients are derived from these binary diffusion coefficients using standard relationships [9].

*F*

^{ij}is a cofactor of

*F*

_{ij}, defined in terms of binary diffusion coefficients as [9]:

_{ij}and

*n*

_{i}correspond to mass density, Kronecker delta symbol and number density of species i. General ambipolar diffusion coefficients are defined in terms of general diffusion coefficients as following [36]:

*Z*

_{j}refer to the number of charges of species j. The total number of ambipolar diffusion coefficients and binary diffusion coefficients computed this way are twenty five each for this problem.

*q*

^{ij}elements in terms of cross-sections,

*Q*

_{ij}

^{(l,s)}, have been used in the computation.

*q*

_{ij}

^{mp}block actually presents an array as following:

*q*

_{ij}

^{(l,s)}elements, necessary to evaluate the above determinant have been computed according to Devoto[30].

### Thermal Conductivity of Electrons

### Thermal Conductivity of Heavy Particles

### Electrical Conductivity

*q*

^{ij}elements as obtained in the diffusion expressions as follows [11]:

### Viscosity

_{2}, O, O

^{+}and O

^{++}. In this study viscosity is computed using first approximation as follows [30]:

### Reactive Thermal Conductivity of Electrons and Heavy Particles

*p*

_{j}for species j, the particle flux vector of species s is given by:

_{1}, Δ h

_{2}and Δ h

_{3}are respectively the heat of reactions specified in Eqs. (1), (2) and (3) and can be estimated as;

*E*

_{d},

*E*

_{1}and

*E*

_{2}are respectively, dissociation, first and second ionization energies.

*Q*

_{int}refers to the internal partition function. Θ

_{rot}and Θ

_{vib}are respectively characteristic rotational (=2.09 K for O

_{2}) and vibrational (=2,260 K for O

_{2}) temperatures.

*T*

_{e}, while the second one appears in the heavy particle energy equation and is associated with ∇

*T*

_{h}. The reactive thermal conductivities are finally computed after evaluating the derivatives employing Eqs. (4)–(6).

### Chemical and Thermal Non-equilibrium

_{k}is the diffusive flux of species ‘k’. The diffusive flux can be written as [36] :

*E*

^{X}is the external electric field.

*S*

_{2}is the equilibrium rate constant of reaction-2 and α

_{I}is the recombination coefficients of ions. Equation (35) can now be written in the form:

*R*

_{2}is the ultimate rate constant for reaction-2 (atom to ion formation) obtained through a modification of the equilibrium rate constant S

_{2}by chemical non-equilibrium contribution Δ

*S*

_{2}.

Similar rate constants *R* _{i}, equilibrium rates *S* _{i} and corresponding non-equilibrium contributions Δ*S* _{i} can be obtained for i^{th} associated reaction.

*r*

_{i}’ for the i

^{th}associated reaction as the ratio:

Similar to the temperature non-equilibrium parameter θ, *r* _{i} can now identify the exact effect of existing chemical non-equilibrium for the i^{th} reaction. Results are presented estimating such effects at different pressures and thermal non-equilibria.

## Results

Using the above method of computation, results obtained are presented in the following. Particular θ values: 1, 2, 3, 5, 10, 20 and 30 are chosen judiciously to maintain clarity of presentation and ranges of interest. Chosen pressure values for presentation are 0.1, 1.0, 4.0 and 7.0 atmospheres. Selected higher and lower pressure ranges are close to respectively upstream and downstream pressures inside of arc plasma torches. Although, value of effective rate constant may exceed corresponding equilibrium rate constant by orders of magnitude depending on plasma condition and geometry, values of ‘*r*’ in the present computation are chosen to range from 0.5 to 2. This is based on a non-equilibrium modeling study of an oxygen cutting torch (2 mm nozzle diameter) where ‘r’ for dominant reactions are found to range from 0.7 to 1.5 within the computational domain. Results of chemical non-equilibria are presented accordingly. Overall property behavior differs from that in monatomic gases mainly due to the presence of dissociative reactions at lower heavy species temperatures. Thermodynamic properties are presented first. Results of transport properties are presented subsequently starting with a comparison with published results under equilibrium. Unless otherwise specified, the numbers associated with each curve in the presented results indicate corresponding θ values. All results presented up to Fig. 15 assume *r* _{i}=1 for all reactions. In an effort to illustrate the possible effects of the chemical-non-equilibrium (*r* _{i} ≠ 1) on the properties, the graphs [Figs. 16–19] are presented with *r* _{1} = *r* _{2} = *r* _{3} = *r*, where *r* varies from 0.5 to 2. It must be emphasized that the three non-equilibrium parameters (*r* _{1}, *r* _{2} and *r* _{3}) are very likely to differ from each other in most of the cases. The parameters are chosen equal just to limit the number of graphs in the presentations. Possible formations of negative ions due to attachment of electrons are completely neglected in the present computation in all cases. At lower pressure (≤ 1atm) presence of O^{3+} may significantly influence the property at temperatures beyond 40,000 K due to enhanced Coulomb cross-section of O^{3+}. While the computed results for higher pressures may be reasonably accurate, at lower pressure, due to omission of O^{3+}, they may be inaccurate by more that 5% for temperatures above about 45,000 K for 1 atm and for temperatures above about 40,000 K for 0.1 atm.

*p*,

*T*

_{e}and θ. Although, there is a strong dependence of the total number density on pressure, the trend of variation with θ remains almost identical. It can be noted that the peak in number density occurs around

*T*

_{e}equal to 15,000K for θ up to around 5 and then the peak decreases and also shifts in position towards higher

*T*

_{e}. For θ > 5 and

*T*

_{e}around 15,000 K,

*T*

_{h}will be below the dissociation temperature and the electron number density (n

_{e}) becomes very small, because there is almost no dissociation and therefore, no ionization. For θ ≥ 10, the peaks of n

_{e}shift to higher values of

*T*

_{e}, because dissociation and hence ionization are delayed until

*T*

_{h}(=

*T*

_{e}/10) reaches dissociation temperature. Dissociation of molecules serves as the root cause for the generation of every other species present in the system.

*T*

_{e}unless some reactions like dissociation or ionization occur at certain temperatures. Similar to earlier observations, the behavior for higher θ are distinctly different from behavior at lower θ caused by similar reasons. Except a drift of the curves toward higher

*T*

_{e}, the overall variation in enthalpy with change of pressure is small. Specific heat at constant pressure (c

_{p}) for electrons and heavy particles are derived from the respective enthalpy component behavior with respect to

*T*

_{e}and

*T*

_{h}and presented in Figs. 5 and 6 respectively. Particular variation of the number density with temperature for a given θ contributes towards the height of the peaks observed.

^{+}collision integrals.

*T*

_{e}.

*T*

_{h}at higher θ for heavy particles result in the gradually increasing size of the peaks with θ for the heavy particle thermal conductivity.

*T*

_{e}, if θ increases,

*T*

_{h}decreases, resulting in a drop in the momentum transport and hence a drop in the viscosity value. With increasing θ, the peaks in the viscosity shift towards higher

*T*

_{e}values. The viscosity increases with increasing pressure.

Volumetric collision frequencies between electrons and heavy particles are presented in Fig. 15. Decrease in collision frequency with electron temperature is primarily caused by a decrease in number density of electrons with temperature. The shift of the peaks can be explained again by the behavior of the dissociation temperature for a given θ.

*r*

_{1}=

*r*

_{2}=

*r*

_{3}=

*r*.

*r*= 1 corresponds to chemical equilibrium. Figures 16 and 17 present results at pressures of 1 and 7 atmospheres respectively for θ = 1. It is observed that at lower temperatures, deviations due to chemical non-equilibrium are relatively small. However, at higher temperatures, these deviations are appreciable. A similar scenario is observed in the properties presented in Figs. 18 and Fig. 19 for θ = 5 and similar pressures. The effects of variations due to θ are somewhat more pronounced for the viscosity.

## Summary And Conclusions

In this paper, values for thermodynamic and transport properties are presented for an oxygen plasma under non-equilibrium conditions. The non-equilibrium properties have been calculated for the conditions of a two-temperature plasma, with an electron temperature different from that of the heavy species temperature, and for a plasma where there is in addition a deviation from composition equilibrium due to diffusion or delayed recombination effects. The property values are given for a range of pressures of 0.1 to 7 atm, and for a range of electron temperatures from 300 K to 45,000 K. This range is wider than the ranges usually found in the literature because it was found necessary to provide values for CFD calculations where there may be an oscillatory behavior of the solution before convergence. The non-equilibrium considered is expressed as the ratio θ of electron temperature to heavy species temperature, and of the ratio *r* of a species actual density derived from kinetic calculations and that derived for equilibrium at *T* _{e}. These non-equilibrium parameters have been varied from 1 < θ < 30 and 0.5 < *r* < 2. This range of chemical composition non-equilibrium covers most of the situations encountered in practice, in particular if one considers that higher θ values frequently accompany diffusion effects, thus reducing the value of *r* that is observed. While it has been observed that θ has a strong influence on all the properties over a wide range of temperatures, chemical composition non-equilibrium is found to significantly influence the properties only at higher values of electron temperature. The fact that for appropriate values of θ dissociation reactions may co-exist with ionization reactions at high *T* _{e} plays a key role in influencing the property values. This is particularly noticeable in the values for the non-equilibrium specific heats and for the reactive thermal conductivity. Under chemical composition non-equilibrium conditions, the production rates of various species are modified by particle fluxes due to density gradients, and this effect can ultimately be reduced to the use of modified rate constants in the associated rate equations. Results of calculations for θ = 1 and *r* = 1 represent equilibrium values of the properties, and the data obtained for these conditions compare favorably with those published previously.

## Notes

### Acknowledgements

The authors would like to thank Hypertherm Inc. for financial support. One of the authors (S. Ghorui) is thankful to Department of Atomic Energy, India for grant of leave for post-doctoral study.

## References

- 1.Devoto RS (1967) Phys Fluids 10:354CrossRefADSGoogle Scholar
- 2.Devoto RS (1973) Phys Fluids 16:616CrossRefADSGoogle Scholar
- 3.Murphy AB, Arundell CJ (1994) Plasma Chem Plasma Process 14:451CrossRefGoogle Scholar
- 4.Murphy AB (1995) Plasma Chem Plasma Process 15:279CrossRefMathSciNetGoogle Scholar
- 5.Murphy AB (1997) IEEE Trans Plas Sci 25:809CrossRefMathSciNetADSGoogle Scholar
- 6.Murphy AB (2000) Plasma Chem Plasma Process 20:279CrossRefGoogle Scholar
- 7.Fauchais P, Elchinger MF, Aubreton J (2000) J High Temp Material Process 4:21Google Scholar
- 8.Aubreton J, Elchinger MF, Fauchais P, Rat V, Andre P (2004) J Phys D Appl Phys 37:2232CrossRefADSGoogle Scholar
- 9.Hirschfelder JO, Kurtis CF, Bird RB (1964) Molecular theory of gases and liquids, 2nd edn. Wiley, New YorkGoogle Scholar
- 10.Chapman S, Cowling TG (1972) The mathematical theory of transport processes in gases. North-Holland, AmsterdamGoogle Scholar
- 11.Devoto RS (1967) Phys Fluids 10:2105CrossRefADSGoogle Scholar
- 12.Devoto RS (1965) Ph.D. thesis, Stanford UniversityGoogle Scholar
- 13.Miller EJ, Sandler SI (1973) Phys Fluids 16:491CrossRefADSGoogle Scholar
- 14.Kannappan D, Bose TK (1977) Phys Fluids 20:1668CrossRefADSGoogle Scholar
- 15.Bonnefoi C (1983) State thesis, University of Limoges, FranceGoogle Scholar
- 16.Aubreton J, Bonnefoi C, Mexmain JM (1986) Rev Phys Appl 21:365Google Scholar
- 17.Rat V, Andre P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2001) Phys Rev E 64:064091CrossRefGoogle Scholar
- 18.Rat V, Aubreton J, Elchinger MF, Fauchais P (2001) Plasma Chem Plasma Process 21:355CrossRefGoogle Scholar
- 19.Rat V, Andre P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2002) Plasma Chem Plasma Process 22:453CrossRefGoogle Scholar
- 20.Rat V, Andre P, Aubreton J, Elchinger MF, Fauchais P, Lefort A (2002) Plasma Chem Plasma Process 22:475CrossRefGoogle Scholar
- 21.Rat V, Andre P, Aubreton J, Elchinger MF, Fauchais P, Vacher D (2002) J Phys D Appl Phys 35:981CrossRefADSGoogle Scholar
- 22.Aubreton J, Elchinger MF, Rat V, Fauchais P (2004) J Phys D Appl Phys 37:34CrossRefADSGoogle Scholar
- 23.Ramshaw JD (1993) J Non-Equilib Thermodyn 18:121zbMATHCrossRefGoogle Scholar
- 24.Ramshaw JD (1996) J Non-Equilib Thermodyn 21:233zbMATHGoogle Scholar
- 25.Andre P, Aubreton J, Elchinger MF, Rat V, Fauchais P, Lefort A, Murphy AB (2004) Plasma Chem Plasma Process 24:435CrossRefGoogle Scholar
- 26.Chen X, Han P (1999) J Phys D Appl Phys 32:1711CrossRefADSGoogle Scholar
- 27.van de Sanden MCM, Schram PPJM, Peeters AG, van der Mullen JAM, Kroesen GMW (1989) Phys Rev A 40:5273CrossRefADSGoogle Scholar
- 28.Mitchner M, Kruger CH (1973) Partially ionized gases. Wiley, New YorkGoogle Scholar
- 29.Ferziger JH, Kaper HG (1972) Mathetical theory of transport processes in gases. North Holland, AmsterdamGoogle Scholar
- 30.Devoto RS (1966) Phys Fluids 9:1230CrossRefADSGoogle Scholar
- 31.Levin E, Patridge H, Stallcop JR (1990) J Thermophysics 4:469CrossRefADSGoogle Scholar
- 32.Yun KS, Mason EA (1962) Phys Fluid 5:380CrossRefADSGoogle Scholar
- 33.Stallcop JR, Patridge H, Levin E (1991) J Chem Phys 95:6429CrossRefADSGoogle Scholar
- 34.Devoto RS (1976) Phys Fluid 19:22CrossRefADSGoogle Scholar
- 35.Liboff RI (1959) Phys Fluid 2:40zbMATHCrossRefADSGoogle Scholar
- 36.Murphy AB (1993) Phys Rev E 48:3594CrossRefADSGoogle Scholar
- 37.Li HP, Chen X (2001) Chin Phys Lett 18:547CrossRefADSGoogle Scholar
- 38.Bose TK, Kannappan D, Seeniraj RV (1985) Warme-und Stoffubertragung 19:3CrossRefADSGoogle Scholar
- 39.Krinberg IA (1965) High Temp (USSR) 3:606Google Scholar
- 40.Yos JM (1965) Report RAD TF-65, AVCO Corporation, Wilmington, MassachusettsGoogle Scholar
- 41.Neumann W, Sacklowski U (1968) Beitr Plasma Phys 8:57Google Scholar