Radiative heat transfer is a major heat loss mechanism in thermal plasmas generated during arc flashes/faults in switchgear applications or during high current interruption in low voltage circuit breakers. A common way to calculate the radiation balance is by means of approximate non-gray radiation models like P1 or discrete ordinates (DOM), where the frequency dependent absorption and emission are described in a number of frequency intervals (bands) using a constant absorption coefficient in each band. Current work is focused on finding the optimal number of bands as well as band interval boundaries that provide a reasonable level of accuracy in comparison to a full spectral solution. An optimization procedure has been applied to different SF6 and copper vapor gas mixtures for an assumed temperature profile. Radiation model results using optimized band averaged absorption coefficients as well as spectral values are provided and discussed for the exemplary temperature profile

Gnybida, M.

Narayanan, V. R. T.

Rümpler, Ch.

English

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

doi:10.14311/ppt.2017.1.70Plasma Physics and Technology 4(1):70–73, 2017 © Department of Physics, FEE CTU in Prague, 2017OPTIMAL BAND SELECTION FOR THE CALCULATION OF PLANCKMEAN ABSORPTION COEFFICIENTSM. Gnybidaa,∗, Ch. Rümplerb, V.R.T. Narayananaa Eaton European Innovation Center, Borivojova 2380, 252 63 Roztoky, Czech Republicb Eaton, 1000 Cherrington Pkwy Moon Township, PA 15108, USA∗ MykhailoGnybida@Eaton.comAbstract. Radiative heat transfer is a major heat loss mechanism in thermal plasmas generatedduring arc flashes/faults in switchgear applications or during high current interruption in low voltagecircuit breakers. A common way to calculate the radiation balance is by means of approximate non-grayradiation models like P1 or discrete ordinates (DOM), where the frequency dependent absorptionand emission are described in a number of frequency intervals (bands) using a constant absorptioncoefficient in each band. Current work is focused on finding the optimal number of bands as well asband interval boundaries that provide a reasonable level of accuracy in comparison to a full spectralsolution. An optimization procedure has been applied to different SF6 and copper vapor gas mixturesfor an assumed temperature profile. Radiation model results using optimized band averaged absorptioncoefficients as well as spectral values are provided and discussed for the exemplary temperature profile.Keywords: radiation, thermal plasma, circuit breaker, SF6.1. IntroductionIn various applications in the area of low-, medium-,and high-voltage power distribution, thermal plasmasare either used as switching elements for current in-terruption (e.g. circuit breakers) or arcs can occuras fault conditions (arc flash in switchgear). In orderto enhance the design and performance of these de-vices, modeling approaches that describe the behaviorof thermal plasmas have been developed [1] and en-hanced in order to include more physical phenomenaand cover a broader range of applications [2]. Herebya set of coupled equations for flow and electrodynamicprocesses is usually solved (magneto-hydrodynamic ap-proach), in combination with one of various simplifiedmodels to consider energy transfer due to radiation.The simplest approach to model the effect of radia-tion is the net emission coefficient (NEC) [1], whichdescribes the net radiation from an isothermal sphereor cylinder. But in various applications it is necessaryto model not only the energy sink in the arc coredue to the net emission, but also the reabsorption inthe colder arc fringe and the radiative flux impingingon surfaces needs to be described to achieve highermodel fidelity. Radiative heat flux contributes to theablation of plastic materials, which can substantiallyinfluence the arc behavior [3].Commonly used approximate radiation transfermodels are the P1 and the discrete-ordinates model(DOM) [4], where the radiation spectrum is dividedinto several gray bands. A constant Planck or Rosse-land averaged absorption coefficient is derived fromthe spectral absorption coefficient. Instead of resolvingthe whole spectrum, the radiation transfer is solvedfor a small number of bands only, which does reducethe computational effort dramatically while providingreasonable accuracy.The selection of the number and location of theband limits has an influence on the achievable ac-curacy and it is not obvious how many bands aresufficient and where to place the bands across thespectrum. To address these questions, numerical opti-mization procedures have been developed to find anoptimal number and distribution of the bands [5, 6].In this paper, we apply the optimization procedure asintroduced in [5, 7] to a specific mixture of SF6 andcopper vapor. Since the amount of metal vapor in thearc chamber or switchgear can vary from 0 to 100%in the real application, it is necessary to quantify theerror that is introduced when using the same bandlimits for various mixture ratios. By using the sameband intervals for different mixtures and solving ra-diation transfer for the full spectrum as well as forthe identified bands, the error introduced by the bandselection can be identified.2. Optimization and verificationPlanck averaging is used to calculate the mean ab-sorption coefficient kPlanck,i for each band i accordingto:kPlanck,i =∫ νi+1νikνBνdν∫ νi+1νiBνdν, (1)where kν is the spectral absorption coefficient and Bνrepresents the black body radiative intensity. Thelower and upper interval boundaries (band limits) aredenoted by νi and νi+1 respectively. We renormalizethe line contribution of the spectral absorption coef-ficients according to [8] with a characteristic plasmaabsorption length H = 3 ·Rp, as suggested in [7]. The70vol. 4 no. 1/2017 Optimal band selection for Planck mean absorption coefficientsFigure 1. Assumed temperature profile.Figure 2. Spectral absorption coefficient for a 90% SF6and 10% Cu vapor mixture at 10 000 K and 1 atm.plasma radius Rp is defined as radius value where thetemperature decreased to 50% of the core tempera-ture.In our optimization procedure, we assume an in-finitely long cylindrical calculation domain with 10 cmradius. The domain is evenly filled with a 90% SF6and 10% Cu vapor mixture at 1 atm pressure (all gasmixture proportions are noted as mass fractions inthis paper). This mixture could be exemplary for theinitial phase during an arc flash event in gas insu-lated medium voltage switchgear. The plasma insidethis domain has a temperature distribution along theradius as shown in Fig. 1 with 13 460 K maximumin the arc core and 300 K at the outside boundary.This temperature profile represents a plasma radius of4 cm according above mentioned definition. The equi-librium species composition for this gas mixture andother mixtures of SF6 and copper vapor in the temper-ature range of interest is calculated by minimizationof Gibbs free energy. The species composition is theinput for the calculation of the spectral absorption co-efficients kν in a frequency range that covers 1012 Hzto 1016 Hz with frequency step size of 2 · 1010 Hz.The calculated spectral absorption coefficient for a90% SF6 and 10% Cu vapor mixture at 10 000 K and1 atm is shown in Fig. 2.A baseline calculation of the radiation transfer re-solving the whole spectrum is the first step in theoptimization procedure (spectral solution [5]). Thisis a computationally expensive step because the radi-ation transfer needs to be solved for each of the ap-proximately 500 000 points in the spectrum. A spatialdiscretization of 3.45 mm is used hereby (30 points).In the second step, the radiation transfer is solved fora fixed number of spectral bands using Planck meanabsorption coefficients. The optimization algorithmvaries the band limits in order to minimize the valueof objective function ∆Fν according to Eq. (2):∆Fν =130√√√√ 30∑i=1(∇ · Fexact,i −∇ · Fmean,i)2 , (2)where ∇ · Fexact,i represents the divergence of the ra-diative flux calculated in the spectral solution for eachpoint i and ∇ · Fmean,i represents the divergence ofradiative flux calculated using the line limited Planckmean absorption coefficients. This objective func-tion describes the difference between the spectral andthe band averaged solution, in particular the energysource/sink term due to radiation transfer. Alterna-tively, the radiative intensity could be used to definethe objective function as well [6].Based on the variation of ∆Fν due to varying bandlimits, the procedure is able to determine an optimalset of band limits. Although the algorithm would beable to find a global minimum, we limit the compu-tational time and abort the calculations if a definedmaximum number of iterations is reached. This pro-cedure is repeated for a different number of bands(2 to 6) to analyze the achievable accuracy when usingsmaller or larger number of bands. Since a largernumber of bands increases the computational effortwhen solving radiation transfer in complex three di-mensional CFD models, it is beneficial to use thesmallest number of bands that still provides sufficientaccuracy.The optimization procedure delivers a set of bandlimits for the specified uniform gas mixture, but inrealistic applications any mixture of gas and metalvapor could be present. Thus, we need to quantifythe accuracy of the radiation transfer model usingthe band averaged absorption coefficients for variousother mixtures. For this evaluation, the differencebetween spectral solution and band averaged solutionis calculated using the 2D radiation transfer codeRAT [4], which is enhanced to calculate radiationtransfer resolving the whole spectrum or utilizing bandaveraged coefficients. In this verification step, thesame temperature profile as shown in Fig. 1 is usedwith a spatial resolution of 100×100 points. A relativeerror norm ∆F ′ν is used to quantify the differencebetween the two approaches [5], where the error isevaluated along the radius only:∆F ′ν =√∑i (∇ · Fexact,i −∇ · Fmean,i)2∑i (∇ · Fexact,i)2. (3)71M. Gnybida, Ch. Rümpler, V.R.T. Narayanan Plasma Physics and TechnologyFigure 3. Optimized mean absorption coefficient bandboundaries for 90% SF6 − 10% Cu vapor mixture.Figure 4. Error norm ∆F ′ν for 90% SF6 − 10% Cuvapor mixture.3. ResultsThe optimization procedure results for90% SF6 − 10% Cu vapor mixture at 1 atm areshown in Fig. 3. The derived band boundaries forthe different total number of bands are all locatedin UV region of the spectrum. In case 2 to 4 bandnumbers, the first interval boundary is located ataround 116.5 nm. Also, in case of 5 or 6 bands, thesame interval boundary is selected, but more intervalsare introduced below and above that wavelength.Comparing the band boundaries with spectralabsorption coefficient as shown in Fig. 2, the 116.5 nminterval boundary can be identified as larger changein the continuum contribution. The other intervalboundaries cannot be associated with the spectralabsorption coefficient that distinctively.Corresponding error norm values according to Eq. 2are provided in Fig. 4. The 2–band approximation hasan error of 13.6 % which is decreased with increasingnumber of bands down to 4.4 % for the 6–band approx-imation. A larger number of bands does improve theaccuracy of the radiation model as expected, but thedifference between 5–band and 6–band approximationis only about 0.3 %. For this specific gas mixture andtemperature profile, solving a 6–band model wouldnot provide a benefit over a 5–band model.The influence of copper vapor on the radiation trans-Figure 5. Divergence of radiative flux for various gasmixtures, spectral resolved model.Band 90%SF6 50%SF6 100% 100%10%Cu 50%Cu SF6 Cu6 band 4.4% 13.6% 12.0% 14.5%3 band 8.1% 14.9% 11.9% 16.4%Table 1. Error norm values for different mixtures andband approximations at 1 atm.fer can be derived from Fig. 5. The divergence ofthe radiative flux along the domain radius as resultof the spectrally resolved model is shown for thesegases/gas mixtures: 100% SF6, 90% SF6 − 10% Cu,50% SF6 − 50% Cu, and 100% Cu. The divergence ofradiative flux in the arc core is increased with largercopper vapor fraction. Compared to pure SF6, thecooling term is increased by an order of magnitudefor pure copper vapor, due to the presence of strongmetallic lines in the spectrum.The influence of different mixtures on the accuracyof the band averaged models is depicted in Fig. 6 forthe same four gases/gas mixtures at 1 atm. Spec-trally resolved model results as well as the 3–bandand 6–band model results are provided, where bandaveraging is done using band limits according to Fig. 3(optimized for 90% SF6 − 10% Cu vapor mixture). Incase of pure SF6, the 3–band model overpredicts theemission by 13.7% (∇ · F in the arc core), althoughthe reabsorption in the fringe is represented well. Thiscan lead to an overestimation of the radiation losses inan arc model later on. But in case of 50% copper va-por admixture, an underprediction of the reabsorptionin the fringe of the temperature profile is noticeable.This can influence the temperature profile as result ofan arc model. The error values as provided in Tab. 1can be used to quantify the accuracy for the wholeprofile.The 90% SF6 − 10% Cu vapor mixture and 6–bandapproximation has the lowest error value of 4.4%, be-cause this mixture was used to find optimal bandboundaries. The error value has a maximum of 16.4%in case of pure copper vapor and 3–band approxi-mation. Reducing the number of bands from 6 to 372vol. 4 no. 1/2017 Optimal band selection for Planck mean absorption coefficients(a) 90% SF6 − 10% Cu (b) 50% SF6 − 50% Cu(c) 100% SF6 (d) 100% CuFigure 6. Divergence of radiative flux for different mixtures at 1 atm.in case of 90% SF6 − 10% Cu vapor mixture almostdoubles the error from 4.4% to 8.1%, but the errorresulting from different gas composition is larger. Inthis case, a larger number of bands does help to limitthe error if gas mixture is varied as well.4. ConclusionsApproximate radiation transfer models are used in arcmodeling and band averaged absorption coefficientsare necessary model input. We presented a numer-ical approach towards an optimal selection of bandboundaries for the calculation of Planck mean absorp-tion coefficients. The optimization was performedfor a 90% SF6 − 10% Cu vapor mixture and a definedtemperature profile. The comparison between bandaveraged radiation model and spectral model resultsallows the quantification of error introduced by bandaveraging. Based on the results, it is feasible to per-form the optimization for a 90% SF6 − 10% Cu vapormixture and apply the band limits to various mixturesof SF6 and copper, at least for the investigated temper-ature profile. The influence of different temperatureprofiles is a topic for further investigations.AcknowledgementsThe authors thank P. Kloc and M. Bartlova for providingthe radiation database.References[1] A. Gleizes, J. J. Gonzalez, and P. Freton. Thermalplasma modelling. Journal of Physics D: Applied Physics,38(9):R153, 2005. doi:10.1088/0022-3727/38/9/R01.[2] Ch. Rümpler and V.R.T Narayanan. Arc modelingchallenges. Plasma Physics and Technology,2(3):261–270, 2015.[3] Ch. Rümpler, H. Stammberger, and A. Zacharias. Low-voltage arc simulation with out-gassing polymers. In 2011IEEE 57th Holm Conf. on El. Contacts (Holm), pages1–8, Sept 2011. doi:10.1109/HOLM.2011.6034770.[4] Michael F. Modest. Radiative Heat Transfer. Thirdedition. Academic Press, 2013.[5] P. Kloc, V. Aubrecht, M. Bartlova, O. Coufal, and Ch.Rümpler. On the selection of integration intervals forthe calculation of mean absorption coefficients. PlasmaChem Plasma Process, 35:1097–1110, 2015.doi:10.1007/s11090-015-9648-3.[6] L. Fagiano and F. Gati. On the order reduction of theradiative heat transfer model for the simulation of plasmaarcs in switchgear devices. J Quant Spec Rad Trans,169:58–78, 2016. doi:10.1016/j.jqsrt.2015.10.002.[7] P. Kloc, V. Aubrecht, and M. Bartlova. Effectiveplasma radius for planck mean absorption coefficient. InProceedings of 21st International Conference on GasDischarges and their Applications, pages 89–92, 2016.[8] H. Nordborg and A. Iordanidis. Self-consistentradiation based modelling of electric arcs: I. efficientradiation approximations. J Phys D: Appl Phys,41(13):135205, 2008.doi:10.1088/0022-3727/41/13/135205.73

Optimal Band Selection for the Calculation of Planck Mean Absorption Coefficients

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Optimal Band Selection for the Calculation of Planck Mean Absorption Coefficients

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