Acta Polytechnica Acta Polytechnica 53(2):98–102, 2013 © Czech Technical University in Prague, 2013 available online at http://ctn.cvut.cz/ap/ MULTIGROUP APPROXIMATION OF RADIATION TRANSFER IN SF6 ARC PLASMAS Milada Bartlova∗, Vladimir Aubrecht, Nadezhda Bogatyreva, Vladimir Holcman Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 10, 616 00 Brno, Czech Republic ∗ corresponding author: bartlova@feec.vutbr.cz Abstract. The first order of the method of spherical harmonics (P1-approximation) has been used to evaluate the radiation properties of arc plasmas of various mixtures of SF6 and PTFE ((C2F4)n, polytetrafluoroethylene) in the temperature range (1000 ÷ 35 000) K and pressures from 0.5 to 5 MPa. Calculations have been performed for isothermal cylindrical plasma of various radii (0.01 ÷ 10) cm. The frequency dependence of the absorption coefficients has been handled using the Planck and Rosseland averaging methods for several frequency intervals. Results obtained using various means calculated for different choices of frequency intervals are discussed. Keywords: SF6 and PTFE plasmas, radiation transfer, mean absorption coefficients, P1-approximation. 1. Introduction An electric (switching) arc between separated contacts is an integral part of a switching process. For all kinds of high power circuit breakers, the basic mechanism is to extinguish the switching arc at the natural current zero by gas convection. The switching arc is responsible for proper discon- nection of a circuit. In the mid and high voltage region, SF6 self-blast circuit breakers are widely used. Radia- tion transfer is the dominant energy exchange mecha- nism during the high current period of the switching operation. Due to the extreme conditions, experimen- tal work only gives global information instead of local information, which may be important for determining the optimum operating conditions; theoretical mod- elling is then of great importance. Several approxi- mate methods for radiation transfer in arc plasma have been developed (isothermal net emission coefficient method [7, 1, 8], partial characteristics method [2, 11], P1-approximation [9], discrete ordinates method [9], etc.). In this paper, the P1-approximation has been used to predict radiation processes in various mixtures of SF6 and PTFE plasmas. 2. P1-approximation If diffusion of light is neglected and local thermody- namic equilibrium is assumed, the radiation transfer equation can be written as Ω ·∇Iν(r, Ω) = κν(Bν − Iν), (1) where Iν is the spectral intensity of radiation, Ω is a unit direction vector, κν is the spectral absorption co- efficient, and Bν is the Planck function – the spectral density of equilibrium radiation. In P1-approximation, the angular dependence of the specific intensity is assumed to be represented by the first two terms in a spherical harmonics expansion Iν(r, Ω) = c 4π Uν(r) + 3 4π Fν(r) · Ω , (2) where Uν denotes the radiation field density, Fν is the radiation flux, and c is the speed of light. Combin- ing this expression with Eq. 1, one finds for radiation flux Fν(r) = − c 3κν ∇Uν(r) (3) and a simple elliptic partial differential equation for the density of radiation Uν ∇· [ −c 3κν(T) ∇Uν(r) ] +κν(T)cUν(r) = κν(T)4πBν(T) . (4) Integrating over frequency, the total density of the ra- diation and the total radiation flux are obtained U(r) = ∫ ∞ 0 Uν(r) dν, F(r) = ∫ ∞ 0 Fν(r) dν. (5) 3. Absorption coefficients Prediction of both radiation emission and absorp- tion properties requires knowledge of the spectral coefficients κν of absorption as a function of radia- tion frequency. These coefficients are proportional to the concentration of the chemical species occur- ring in the plasma, and depend on the cross sections of various radiation processes. In the mixture of SF6 and PTFE (C2F4) we assume the following species: SF6 molecules, S, F, C atoms, 98 http://ctn.cvut.cz/ap/ vol. 53 no. 2/2013 Multigroup Approximation of Radiation Transfer in SF6 Arc Plasmas S+, S+2, S+3, F+, F+2, C+, C+2, C+3 ions and elec- trons. The equilibrium concentrations of each species in various SF6 + PTFE mixtures were taken from [3]. Spectral coefficients of absorption were calculated using semi-empirical formulas to represent both con- tinuum and line radiation. The continuum spec- trum is formed by bound-free transitions (photo- recombination, photo-ionization) and free-free tran- sitions (bremsstrahlung). The photo-ionization cross sections for neutral atoms were calculated by the quan- tum defect method of Seaton [12], the cross sections of the photo-ionization of ions and free-free tran- sitions were treated using Coulomb approximation for hydrogen-like species [6]. In the discrete radiation calculations, spectral lines broadening and their com- plex shapes have to be carefully considered. The lines are broadened due to numerous phenomena. The most important are Doppler broadening, Stark broadening, and resonance broadening. For each line, we have cal- culated the values of half-widths and spectral shifts. The line shape is given by convolution of the Doppler and Lorentz profiles, resulting in a simplified Voigt profile. The lines that overlap have also been taken into account. Due to lack of data, from molecular species we have only considered SF6 molecules with their experimentally measured absorption cross sec- tions [5]. 4. Absorption means One of the procedures for handling the frequency vari- able in the radiation transfer equation is the multi- group method [10, 4]. It is based on a simplified spectral description with only some spectral groups assuming grey body conditions within each group with a certain average absorption coefficient value, i.e. for the k-th spectral group κν(r,ν,T) = κk(r,T); νk ≤ ν ≤ νk+1. (6) The mean absorption coefficient values are generally taken as either the Rosseland mean or the Planck mean. The Planck mean is appropriate in the case of an op- tically thin system. The Planck mean absorption coefficient is given by κP = ∫νk+1 νk κνBν dν Bk , (7) where Bk = ∫ νk+1 νk Bν dν. The Rosseland mean is appropriate when the sys- tem approaches equilibrium (almost all radiation is reabsorbed). The Rosseland mean is given by κ−1R = ∫νk+1 νk κ−1ν dBν dT dν∫νk+1 νk dBν dT dν . (8) The total radiation density value is then given by U(r) = ∑ k Uk(r), (9) where Uk are solutions of Eq. 4 with frequency inde- pendent κk(T) and Bk(T). 5. Net emission coefficients Assuming local thermodynamic equilibrium, coeffi- cient of absorption κν is related to the coefficient of emission εν by Kirchhoff’s law εν = Bνκν. (10) Strong self-absorption of radiation in the plasma volume occurs, and this must be taken into account in the calculations. The net emission coefficient of ra- diation, εNν is defined by Lowke [7] as εNν = εν −Jνκν , (11) where Jν is an average radiation intensity, which is a function of temperature. For an isothermal plasma sphere at radius R (the results are approximately the same as for the isothermal cylinder), it is defined as Jν = Bν[1 − exp(−κνR)]. (12) A combination of Eqs. 10–12 gives the expression for the net emission coefficient εN = ∫ ∞ 0 Bνκνexp(−κνR) dν. (13) The isothermal net emission coefficient corresponds to the fraction of the total power per unit volume and unit solid angle irradiated into a volume surrounding the axis of the arc plasma and escaping from the arc column after crossing thickness R of the isothermal plasma. It is often used for predicting the energy balance, since the net emission of radiation (the di- vergence of the radiation flux) can be written as ∇· FR = 4πεN. (14) In multigroup P1-approximation, the net emission coefficient can be determined from Eq. 4. In the case of cylindrically symmetrical isothermal plasma, Eq. 4 has constant coefficients κk and Bk, and depends only on one variable – radial distance r. It represents the modified Bessel equation, and can be solved ana- lytically. Taking into account the boundary condition (no radiation enters into the plasma cylinder from outside) n · Fk(R) = − cUk(R) 2 (15) the net emission over the volume of the arc for the k-th frequency group is (wavg)k = 2π πR2 ∫ R 0 r∇· Fk(r) dr = = 2 R 4πBk 2I1( √ 3 κkR) + √ 3 I0( √ 3 κkR) I1( √ 3 κkR) (16) 99 M. Bartlova, V. Aubrecht, N. Bogatyreva, V. Holcman Acta Polytechnica 2 4 6 8 10 10-4 10-2 100 102 104 A bs or pt io n C oe ffi ci en t ( cm -1 ) Radiation Frequency (1015 s-1) (a) Absorption Spectrum Planck Mean Rosseland Mean SF 6 p = 0.5 MPa T = 20 000 K Figure 1. The real absorption spectrum of SF6 plasma at p = 0.5 MPa and T = 20 000 K compared with the Planck and Rosseland means. 10000 20000 30000 10-3 10-1 101 103 105 107 SF 6 p = 0.5 MPa R = 0.1 cm N et E m is si on C oe ffi ci en t ( W cm -3 sr -1 ) Temperature (K) Planck Mean, 5 gr. Rosseland Mean, 5 gr. Planck mean, 10 gr. Rosseland Mean, 10 gr. Aubrecht [2] Figure 2. Net emission coefficients of SF6 plasma with radius 0.1 cm as a function of temperature for two different cuttings of the frequency interval and various absorption means; comparison with re- sults of Aubrecht [1]. where I0(x) and I1(x) are modified Bessel functions. Summing over all frequency groups gives the net emis- sion of radiation ∇· FR = ∑ k (wavg)k = 4πεN. (17) 6. Results The mean absorption coefficient values depend on the choice of the frequency interval cutting. 5000 10000 15000 20000 25000 30000 35000 10-1 100 101 102 103 104 105 106 N et E m is si on C oe ffi ci en t ( W cm -3 sr -1 ) Temperature (K) Aubrecht [2] Planck Mean Rosseland Mean R = 0 R = 1 cm R = 10 cm SF 6 p = 0.5 MPa Figure 3. Net emission coefficients of SF6 plasma as a function of temperature for various thicknesses of the plasma and various absorption means. 5000 10000 15000 20000 25000 30000 35000 10-1 101 103 105 107 N et E m is si on C oe ffi ci en t ( W cm -3 sr -1 ) Temperature (K) p = 0.5 MPa p = 2 MPa p = 5 MPa SF 6 R = 0.1 cm Planck Rosseland Figure 4. Net emission coefficients of SF6 plasma with radius 0.1 cm as a function of temperature for var- ious pressures. The cutting frequencies are mainly defined by the steep jumps of the evolution of the continuum absorp- tion coefficients that correspond to individual absorp- tion edges. However, the number of groups should be minimized to decrease the computation time. In this work, the frequency interval (1012 − 1016) s−1 was cut into (a) five frequency groups with cutting frequencies (0.001, 1, 2, 4.1, 6.8, 10) × 1015 s−1 , (18) (b) ten frequency groups with cutting frequencies (0.001, 1, 1.4, 1.77, 2, 2.2, 2.5, 3, 4.1, 6.8, 10) × 1015 s−1 . (19) 100 vol. 53 no. 2/2013 Multigroup Approximation of Radiation Transfer in SF6 Arc Plasmas 10000 20000 30000 101 102 103 104 105 106 N et E m is si on C oe ffi ci en t ( W cm -3 sr -1 ) Temperature (K) 100% SF 6 80% SF 6 + 20% PTFE 20% SF 6 + 80% PTFE 100% PTFE p = 0.5 MPa R = 0.1 cm Planck Rosseland Figure 5. Net emission coefficients of different mix- tures of SF6 and PTFE plasmas as a function of tem- perature at pressure 0.5 MPa for various absorption means. The two cuttings differ in the frequency interval (1 ÷ 4.1) × 1015 s−1, which is split into two groups in case (a), and in greater detail into seven groups in case (b). The absorption spectrum evaluated at 20 000 K compared with various averaged versions for five groups cutting is shown in Fig. 1. The net emission coefficients were calculated by combining Eqs. 16 and 17. Results for an isothermal plasma cylinder of radius R = 0.1 cm for two different cuttings Eqs. 18, 19 of the frequency interval are given in Fig. 2. In the case of the Planck averaging method, cutting the spectrum into more frequency groups in- fluences the resulting net emission coefficients only slightly. A comparison is also provided with the values of Aubrecht [1], which were obtained by direct integra- tion from Eq. 13. It can be seen that the Planck mean leads to an overestimation of the emitted radiation, while the Rosseland approach underestimates it. An example of the calculated temperature depen- dence of the net emission coefficients for various thick- nesses of pure SF6 plasma at a pressure of 0.5 MPa is presented in Fig. 3. The strong effect of plasma thickness can be seen both for direct frequency integra- tion Eq. 13 and for Planck means; Rosseland averages are influenced only slightly. As can be expected from the definition of the Planck and Rosseland means, by omitting self-absorption (R = 0) the Planck means give good agreement with the results of direct integra- tion, while for thick plasma (R = 10 cm) the Rosseland mean is a good approach. The influence of the plasma pressure on the net emission coefficient values is shown in Fig. 4. Net emission coefficients increase with increasing pressure, mainly for Rosseland means. The influence of an admixture of PTFE on the val- ues of the net emission coefficients of SF6 plasma is given in the Fig. 5 for plasma thickness 0.1 cm. The differences between net emission coefficients are very small. This can be explained by the approxi- mately equivalent role of sulphur and carbon species. Sulphur and carbon atoms and ions have similar radi- ation emission behavior. 7. Conclusions Net emission coefficients for various mixtures of SF6 and PTFE plasmas have been calculated using P1- approximation for an isothermal plasma cylinder. Multigroup approximation for handling the frequency variable has been used. Both Planck and Rosseland averaging methods have been applied to obtain mean values of absorption coefficient values. A comparison with the net emission coefficients calculated by di- rect frequency integration has been provided. It has been shown that Planck means generally overestimate the emission of radiation, while Rosseland means un- derestimate it. Planck means give good results only for a very small plasma radius (omitting self-absorption). The Rosseland mean is a suitable approach for thick plasma (absorption dominated system). In reality, neither mean is correct in general. The simplest pro- cedure for improving the accuracy is to use the Planck mean for frequency groups with low absorption coeffi- cient values and the Rosseland mean for groups with high absorption coefficient values. Another approach was suggested in [9], where each group based on original frequency splitting was fur- ther divided according to the absorption coefficient values, and Planck averaging for these new groups was calculated. This procedure partially solves the prob- lem of overestimation of the role of lines in the Planck averaging method. Another correction of the influence of lines on Planck means was presented in [4], where the escape factor was introduced. Acknowledgements This work has been supported by the Czech Science Foundation under project No. GD102/09/H074 and by the European Regional Development Fund under projects Nos. CZ.1.05/2.1.00/01.0014 and CZ.1.07/2.3.00/09.0214. References [1] V. Aubrecht, M. Bartlova. 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Monthly Not Royal Astronom Society 118(5):504–518, 1958. 102 Acta Polytechnica 53(2):98–102, 2013 1 Introduction 2 P1-approximation 3 Absorption coefficients 4 Absorption means 5 Net emission coefficients 6 Results 7 Conclusions Acknowledgements References