In this paper, attention has been given to the absorption properties of the arc plasma at the different pressure conditions. Calculations of the absorption coefficients for a thermal plasma have been performed as a function of the temperature and the frequency. Methods for prediction of the average absorption coefficients were described and compared in detail

Aubrecht, V.

Bartlová, M.

Bogatyreva, N.

Kloc, P.

English

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

doi:10.14311/ppt.2017.3.253Plasma Physics and Technology 4(3):253–256, 2017 © Department of Physics, FEE CTU in Prague, 2017RADIATION TRANSFER IN ARC PLASMASN. Bogatyrevaa,∗, M. Bartlováa, V. Aubrechtb, P. Klocba Department of Physics, Faculty of Electrical Engineering and Communication, Brno University of Technology,Technická 3058/10, 616 00 Brno, Czech Republicb Department of Power Electrical and Electronic Engineering, Faculty of Electrical Engineering andCommunication, Brno University of Technology, Technická 3058/12, 616 00 Brno, Czech Republic∗ bogatyreva@feec.vutbr.czAbstract. In this paper, attention has been given to the absorption properties of the arc plasma atthe different pressure conditions. Calculations of the absorption coefficients for a thermal plasma havebeen performed as a function of the temperature and the frequency. Methods for prediction of theaverage absorption coefficients were described and compared in detail.Keywords: air plasma, radiation transfer, absorption coefficient.1. IntroductionCurrently arc discharges are used increasingly in vari-ous fields of modern science and technology. Certainplasma parameters make its employment very promis-ing for creation of new technologies. Applicationsof the plasma are quite diverse: from equipment forthe modification and processing of polymer materi-als and plasma cutting machines till using plasma inhigh voltage circuit breakers and as well as householdapplications, etc.In most cases optimization of technological condi-tions is the result of experiment during developmentof advanced equipment employing plasma. However,the experimental conditions are rather extreme (hightemperature, pressure and gas velocity). Also, car-rying out experiments is associated with substantialmaterial expenditures. Therefore, it is very importantto develop mathematical approximations and processmodeling of parameters in the arc plasma. Mathe-matical modeling is also quite complex and it requirescertain computational resources, so it is carried outin several stages. Absorption coefficients are primaryparameters in the simulation. Based on that it is pos-sible to define other parameters, which are necessaryfor analyzing of energy transfer in arc plasma.2. Energy balance of arc plasmaElectric discharge plasmas transmit relatively greatdeal of power. The ratios while transmission are de-scribed by the energy balance of the plasma. In thetheory of the energy balance of the plasma only the ele-ment of the plasma volume is taken into considerationand heat convection is neglected.The energy balance of the arc plasma in station-ary state can be described by Ehlenbaas–Heller equa-tion [1]σEE2 = −div [λ · grad(T )] + div(WR), (1)where σEE2 denotes the input electric power (σEis the electric conductivity, E is the electric field),λ · grad(T ) denotes the energy flux due to heat con-duction (λ is the heat conductivity, T is the plasmatemperature, div(WR) denotes the losses of energyby radiation (WR is the radiation flux).Therefore, for modeling of arc plasma includingradiation it is necessary to determine the radiationflux and its divergence. The solution of the equationof radiation transfer makes possible to estimate theseparameters.The complete stationary equation of radiation trans-fer for an absorbing and emitting medium isΩ · grad(Iν) = κν [B(ν)− Iν ], (2)where Ω is the unit direction vector, Iν is the radiationintensity, κν is the absorption coefficient, B(ν) isPlanck’s spectral radiation intensity for equilibriumradiation, ν is the frequency [2].The equation of transfer is obviously very compli-cated. The spectral intensity, which is the dependentvariable in this equation, depends in general uponindependent variables (r, ν,Ω). One must approxi-mate the equation of transfer, either analytically ornumerically, in order to obtain a solution. Due tothe non-linearity of equations describing the radiationfield and strong dependence of input parameters onthe radiation frequency, various approximate methodsare used. One of them is the method of sphericalharmonics – PN -approximation [3–6].The great advantage of the method of sphericalharmonics is the conversion of the equation of trans-fer to relatively simple partial differential equations.However, the low-order approximations are usuallyonly accurate in optically thick media and for higher-order approximations the mathematical complexityincreases rapidly. It is known from neutron transporttheory that approximations of odd orders are moreaccurate than even ones. Due to its simplicity, mainlythe lowest order PN solution corresponding to N = 1253N. Bogatyreva, M. Bartlová, V. Aubrecht, P. Kloc Plasma Physics and Technology(P1-approximation) is usually used (so called diffu-sion approximation). The diffusion approximationdescribes in good accuracy the radiation field in manyproblems of radiation hydrodynamics.In P1-approximation we suppose that the angulardependence of the specific intensity can be representedby the first two terms in a spherical harmonic expan-sionIν(r, ν,Ω) = ϕ1(r, ν) + 3ϕ2(r, ν) ·Ω, (3)where r is the position vector fixing the location of apoint in space, ϕ1 and ϕ2 correspond to the densityof the radiation field multiplied by velocity of light c,and to the radiation flux [7].The spectral density of the radiation field isUν(r, ν) =1c4pi∫0Iν(r, ν,Ω)dΩ =4picϕ1(r, ν). (4)Likewise, for radiation flux we obtainWν = (r, ν)4pi∫0Iν(r, ν,Ω) ·ΩdΩ = 4piϕ2(r, ν). (5)By integration of the equation of radiation transferEq. (2) over all solid angles we obtain the diffusionequation for the spectral density of radiation field−div{c3κνgrad [Uν(r, ν)]}+ κνcUν(r, ν)= κν4piB(ν),(6)and for the radiation fluxWν(r, ν) = − c3κν grad [Uν(r, ν)] . (7)The diffusion approximation is valid under assump-tion that the spectral radiation intensity is almostisotropic.3. Multigroup methodOne of the methods for handling the frequency vari-able in the equation of transfer is the multigroupmethod [8], which leads to its discretization. Oneassigns a given photon to one of G frequency groups,and all photons within a given group are treated thesame from the point of view absorption propertiesof the medium, the absorption coefficient for givenfrequency group k is supposed to be constant withcertain average valueκν(r, ν, T ) = κk(r, T ), νk ≤ ν ≤ νk+1, k = 1, . . . , G.(8)The equation of transfer for the given frequencygroup can be treated as equation for grey medium:Ω · grad [Iν(r,Ω)] = κk νk+1∫νkB(ν)dν − Ik(r,Ω) ,1 ≤ k ≤ G,(9)where κk is the mean absorption coefficient defines asκk =∫ νkνk−1κν(ν) [B(ν)− Iν(r, ν,Ω)] dν∫ νkνk−1[B(ν)− Iν(r, ν,Ω)] dν. (10)For P1-approximation the multigroup diffusion equa-tion has the form−div{c3κkgrad [Uk(r)]}+ κkcUk(r)= κk4piνk∫νk−1B(ν)dν.(11)4. Mean absorption coefficientsFor the multigroup method to be useful, one mustbe able to compute or estimate the mean values ofabsorption coefficients Eq. (10). An exact calculationof these group constants involves knowledge of thespectral intensity I(r, ν,Ω) which is unknown. The un-derlying assumption in the multigroup method is thatthe group constants are relatively insensitive to theweighting functions I(r, ν,Ω) used in computing theaverages over frequency. The accuracy of this methoddepends also on the interval splitting. As the groupwidth approaches zero the group constants becomeindependent upon the estimate made for I(r, ν,Ω).The absorption coefficients are generally complexand widely varying functions of frequency (as canbe seen at Fig. 1. The use of different weightingfunctions I(r, ν,Ω) can lead to the quite differentresults. Generally, κk is taken as either Rosseland orPlanck mean.The Rosseland mean, also called mean free path ofradiation is appropriate when the system approachesequilibrium (almost all radiation is reabsorbed). Ithas the formκ−1R =∫ νkνk−1κ−1νdB(ν, T )dT dν∫ νkνk−1dB(ν, T )dT dν. (12)The Planck mean is appropriate in the case of opti-cally thin and emission dominated system. It can beexpressed asκP =∫ νkνk−1κνB(ν, T )dν∫ νkνk−1B(ν, T )dν. (13)5. Absorption properties of plasmaRadiation in arc plasma depends, besides others physi-cal quantities, on concentrations of all chemical speciesoccurring in the plasma. Equilibrium compositions of254vol. 4 no. 3/2017 Radiation transfer in arc plasmas1 0 2 0 3 0 4 01 02 01 0 2 11 0 2 21 0 2 31 0 2 41 0 2 5  Species, m-3T e m p e r a t u r e ,  k KA i rp  =  0 . 1  M P aO 2A rA r +ON 2e -NN + N + 2N O +N O O +O + 2A r + 2 N + 3A r + 3O + 31 0 2 0 3 0 4 01 02 01 0 2 11 0 2 21 0 2 31 0 2 41 0 2 51 0 2 6  Species, m-3T e m p e r a t u r e ,  k KA i rp  =  1 . 0  M P aO 2N 2Ne -N +O +N + 2O + 2A r +A rA r + 2N + 3A r + 3O + 3N OOO -N O 2N O +N +2N -2N 2 OFigure 1. Concentration of different species in the thermal plasma as a function of temperature.different types of plasma can be computed using somemethods. At our Faculty, a computer code was devel-oped for the calculation of absorption coefficients inthermal plasmas and a database of input parametersfor many types of plasma was created. Here, these pa-rameters were calculated with greater accuracy. Oneof the options considered is clean air without CO2. Inthis case, we take into account availability in plasmasome elements O2, N2, Ar, e− and also ions of O, N,Ar. Figure 1 shows the concentrations of differentspecies in the air plasma for two pressures 0.1 MPaand 1.0 MPa. It is obvious, that the concentrationdepends on the plasma pressure. Therefore valuesof these parameters influence the magnitude of theabsorption coefficients.The absorption coefficients of air plasma werecalculated for eleven different pressures (0.01 MPa,0.1 MPa, 0.2 MPa, 0.4 MPa, 0.6 MPa, 0.8 MPa,1 MPa, 1.5 MPa, 2 MPa, 3 MPa and 4 MPa). Theplasma temperature was considered in the range from300 K to 35 000 K. The frequency interval for calcu-lations was (0.01 to 10) · 1015 Hz with increment of5× 105 Hz. It allows the accuracy increasing of thecalculated data.An example of absorption coefficient for the pressureof 0.1 MPa and temperatures 5000 K and 15 000 K isshown in Fig. 2. This spectrum is a complex of discreteand continuous spectra. High complexity of radiationabsorption coefficients (with respect to the frequencyand temperature) can be seen.Figure 2. Absorption coefficient of air thermal plasmaas a function of frequency.After evaluation of obtained spectra for various pres-sures, expected regularities were revealed for values ofabsorption coefficient. The absorption coefficients atthe pressure of 0.1 MPa and 1.0 MPa were taken forcomparison. The corresponding graphs for tempera-tures of 5000 K and 15 000 K are shown in Fig. 3.Given by the appearance of these spectra, com-puter code was created to analyzing and evaluation ofthe absorption coefficients dependence on frequency.This computer code makes possible to calculate mean255N. Bogatyreva, M. Bartlová, V. Aubrecht, P. Kloc Plasma Physics and TechnologyFigure 3. Comparison of absorption coefficients in air thermal plasma for two pressures.absorption coefficients using methods described pre-viously. Therefore, this software allows to divide thefrequency interval into a certain number of groups.As a result of these procedures, we obtain the meanabsorption coefficients in relation to the temperature.Fig. 4 shows the example of results calculated for twofrequency groups (the whole frequency interval wasdivided into 15 groups).Figure 4. Mean absorption coefficients in air plasmafor two frequency groups (0.76 – 1.42) · 1015Hz and(8.68 – 9.34) · 1015 Hz.6. ConclusionsAnalyzing different ways of dividing into groups anddifferent number of groups, we can conclude thatthese approximation methods give similar results intwo thirds of our frequency interval. In estimatingobtained data and graphs for an eleven different pres-sures we can conclude that the best results yields forRosseland mean in frequency groups with small valuesof absorption coefficient and Planck’s mean with highvalues of absorption coefficient. This occurs due tothe properties of emission and re-absorption. Fromthe comparison follows that Planck means generallyoverestimate the emission of radiation, and Rosselandmeans underestimate it.In accordance with these conclusions we will useat each interval relevant method of obtainment themean absorption coefficients for further calculationand determination of other parameters, which arenecessary for resolve the equation of radiation transfer.Thus, using approximation we can significantly reducecomputational time expenditures, while remaining asufficiently high accuracy of estimations.AcknowledgementsThis research work has been carried out in the Centre forResearch and Utilization of Renewable Energy (CVVOZE).Authors gratefully acknowledge financial support from theMinistry of Education, Youth and Sports of the CzechRepublic under NPU I programme (project No. LO1210)and from the Czech Science foundation under projectNo. GA 15-14829S.References[1] B. Gross and O. Havelka. Elektrické přístroje II.VUT, Brno, 1980.[2] Boulos. Maher I. Thermal plasmas: fundamentals andapplications. 1. Plenum Press, New York, 1994.[3] V. Aubrecht and M. Bartlová. Radiation transfer inair thermal plasmas. In Proc. of XVIth Symposium onPhysics of Switching Arc, pages 9–12, 2005.[4] J.H. Jeans. The equations of radiative transfer ofenergy. Monthly Notices Royal Astronomical Society,78:28–36, 1917. doi:10.1093/mnras/78.1.28.[5] V.G. Sevastyanenko. Radiation transfer in a realspectrum. integration over frequency. J. Eng. Phys.,36:138–148, 1979. doi:10.1007/BF00865111.[6] R.E. Marshak. Note on the spherical harmonics methodas applied to the milne problem for a sphere. Phys. Rev,71:443–446, 1947. doi:10.1103/PhysRev.71.443.[7] B. N. Chetverushkin. Matematicheskoe modelirovaniezadach dinamiki izluchajuschego gaza. Nauka, Moskva,1985.[8] G.C. Pomraning. The equations of radiationhydrodynamics. Dover Publications, New York, 2005.256

Radiation Transfer in Arc Plasmas

https://ojs.cvut.cz/ojs/index.php/PPT/article/download/6225/5399

Radiation Transfer in Arc Plasmas

Abstract

Similar works

Full text

Available Versions

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)