In this paper we present a process for mean absorption coefficient optimal band selection applied to the 3-band model of radiation in an air electrical arc. For fixed temperature, the divergence of radiation flux in an infinite cylindrical plasma column is calculated using spectrally resolved absorption coefficient and serves as a reference value. Optimization process is used to properly select the bands of 3-band mean absorption approximation using both unmodified and limited Planck mean absorption coefficient. The ac-curacy of aforementioned two approximation methods is evaluated

Aubrecht, V.

Bartlova, M.

Coufal, O.

Kloc, P.

English

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

Kloc P. et al.: Optimization of 3-band Mean Absorption Coefficients  Plasma Physics and Technology 2015, 2, 2, 146-149 Optimization of 3-band Mean Absorption Coefficients  Kloc P.1, Aubrecht V.1, Bartlova M.1, Coufal O.1 1Centre for Research and Utilization of Renewable Energy, FEEC, Brno University of Technology, Technicka 10, Brno, Czech Republic, klocpetr@feec.vutbr.cz  In this paper we present a process for mean absorption coefficient optimal band selection applied to the 3-band model of radiation in an air electrical arc. For fixed temperature, the divergence of radiation flux in an infinite cylindrical plasma column is calculated using spectrally resolved absorption coefficient and serves as a reference value. Optimization process is used to properly select the bands of 3-band mean absorption approximation using both unmodified and limited Planck mean absorption coefficient. The ac-curacy of aforementioned two approximation methods is evaluated.  Keywords: radiation transfer, mean absorption coefficient, Planck mean, divergence of radiation flux1 INTRODUCTION Radiation transfer is one of the key factors con-tributing to the energy balance of an electrical arc. Accurate evaluation of energy losses due to radiation transfer is therefore critical for de-velopment of a new electrical devices including switchgears and circuit breakers. However a very complex structure of the absorption co-efficient, often requiring over 100000 fre-quency points to proper evaluate [1], makes any calculation using this exact absorption coeffi-cient computationally unfeasibly demanding. A standard approach to reduce the computa-tional demands is the usage of the mean absorp-tion coefficients (MACs) [1-4]. The entire fre-quency range is divided into several bands and the radiation transfer inside each band is evalu-ated using grey approximation. This approach significantly reduces the computational de-mands at the cost of accuracy. Number of bands 𝑁 is determined solely by the available compu-tational power and required accuracy. The value of mean absorption coefficient inside each band should reflect the general behavior of the spectral absorption coefficient in that particular frequency range. The value of spec-tral absorption coefficient should also not vary extensively. Three different mean values are generally proposed in the literature [1,2]:   Natural mean   𝜅𝑛 =  ∫ 𝜅𝜈𝑑𝜈𝜈2𝜈1𝜈2 − 𝜈1, (1)  Planck mean   𝜅𝑃 =  1∫ 𝐵𝜈d𝜈𝜈2𝜈1∫ 𝜅𝜈𝐵𝜈d𝜈𝜈2𝜈1, (2)  Rosseland mean  1𝜅𝑅= ∫1𝜅𝜈d𝐵𝜈d𝑇d𝜈𝜈2𝜈1∫d𝐵𝜈d𝑇d𝜈𝜈2𝜈1⁄ , (3) where 𝜈1 and 𝜈2 represents the band boundary frequencies, 𝜅𝜈 is a spectral absorption coeffi-cient and 𝐵𝜈 is blackbody radiation. Neither of these mean methods reflect the absorption co-efficient precisely in the entire range. The Planck mean is known to behave well for low values of 𝜅𝜈 whereas Rosseland mean is best suited for high values of 𝜅𝜈 and is thus best suited for strong lines [1]. To compensate for the shortcoming of the Planck mean absorption coefficients the use of escape factors [1] or line limiting factors [2,4] are proposed in literature leading to accuracy enhancement. Another important parameter is the selection of the band boundaries. Given the complexity of the spectral absorption coefficient this selection is very complicated one and fairly unintuitive. Personal experience [4] and the position of large changes in absorption continuum [1] are both used in the literature. In this proceedings we discuss the application of numerical optimi-zation process on the selection of band bound-aries for 3-band mean absorption approxima-tion. Even with this extremely small number of bands, fairly good accuracy can be achieved with proper treatment of the band boundaries. 2 MODEL DESCRIPTION The following model was used to evaluate the accuracy of band boundaries selection and us-ing different mean absorption coefficients. We considered infinite cylindrical plasma column Kloc P. et al.: Optimization of 3-band Mean Absorption Coefficients  147 with fixed axisymetrical temperature profile as shown in Fig. 1.    The properties of the gas inside the column was considered to be similar to the dry air. Ex-ample of calculated spectral absorption coeffi-cient using 245000 spectral points is shown in Fig.2. More detailed information regarding the datasets used to calculate the spectral absorp-tion coefficient and quadrature to calculate the divergence of radiation flux 𝛻𝑭𝜈 in 20 points along the cylinder radius can be found in [5]. We used the same calculation method to calcu-late the divergence of radiation flux 𝛻𝑭𝑀𝐴𝐶  for the 3-band MAC approximation. The bounda-ries between the bands can be specified freely, however they are considered constant in the whole temperature range. Two methods were used to calculate the MAC inside each band. The first method is a classical Planck MAC as defined in (2). This approximation is referred as pure-Planck and denoted by index 𝑃. The second method uses line limiting factor   𝜅(𝑟𝑝) =  1𝑟𝑝(1 − 𝑒−𝜅lines𝑟𝑝) (4) introduced by Nordborg [2]. Based on our ex-perience we chose 𝑟𝑝 = 15mm which seems to provide reasonable level of accuracy. The MAC is then calculated as  𝜅𝑁 =  1∫ 𝐵𝜈d𝜈𝜈2𝜈1∫ 𝜅mod𝐵𝜈d𝜈𝜈2𝜈1, (5) where  𝜅mod =  𝜅cont + 𝜅(𝑟𝑝). (5) This approximation is referred as Nordborg-Planck and denoted by index 𝑃. To evaluate the accuracy of the 3-band MAC approximation, we defined the norm of diver-gence of radiation flux as  𝛥𝐹 = ∑(𝛻𝑭𝜈,𝑖 − 𝛻𝑭𝑀𝐴𝐶,𝑖)2,20𝑖=1 (6) where index 𝑖 denotes the point along the cyl-inder radius. The norm in (6) also represents the objective function which will be minimized with respect to the band boundary positions. The numerical optimization procedure was car-ried out using low dimensional simplex method according to [6]. Since this method is capable locating the global minimum of the objective function, the resulting band boundaries can be considered the optimal ones. To ensure this, the optimization procedure was run several times. 3 RESULTS AND DISCUSSION The optimal positions of band boundary fre-quencies obtained by numerical optimization are shown in Fig.2. In the case of pure-Planck MAC approximation the boundaries are located at 𝜈1 =  2.761 × 1015Hz and 𝜈2 =  2.766 ×1015Hz. This creates a very narrow middle band, which contains several strong resonance lines belonging to N+. These lines are not pre-sent at lower temperatures due to low concen-tration of nitrogen ions. With increasing tem-perature the importance of these resonance lines increases (see Fig.2) to the limit, where they justify the designation of a separate band. This is also enhanced by the fact that Planck mean is known to overestimate the importance of the strong resonance lines.  Fig.2: Mean absorption coefficient boundaries and value compared to realistic absorption coefficient at 25 kK Fig. 1: Assumed temperature profile Kloc P. et al.: Optimization of 3-band Mean Absorption Coefficients  148 The inaccuracy of the 3-band pure-Planck MAC approximation is still extremely large even with careful selection of the band bound-aries. This is clearly demonstrated in Fig.4. The spectral absorption coefficient 𝜅𝜈 contains way too many strong lines for pure-Planck approxi-mation to properly capture the behavior of the spectral absorption coefficient in each band.  Significant improvement can be achieved by using the Nordborg-Planck MAC approxima-tion. By limiting the lines the MACs will more precisely reflect the behavior of the spectral ab-sorption coefficient in each band. In such case it is even more complicated to determine the optimal position of the band boundaries. The optimization procedure positioned the bounda-ries at the frequencies 𝜈1 = 2.938 × 1015Hz and 𝜈2 = 3.513 × 1015Hz. While the position of the second boundary 𝜈2 can clearly be linked to the absorption edge in the continuum no such link can be made for the first boundary 𝜈1 im-plying the necessity for numerical optimization process.  Fig. 5: Radiation flux in radial direction flux using realistic absorption coefficient and two variants of mean absorption coefficients The inclusion of line limiting factors into the MAC calculation indeed provided significant enhancement to accuracy as is shown in Fig.4. The difference between divergence of radiation flux using realistic absorption coefficient and using 3-band Nordborg-Planck approximation is less than 10% in the regions of maximum Fig.3: Temperature dependence of the MACs in each band for 3-band Nordborg-Planck approximation (left) and pure-Planck approximation (right) Fig.4: Divergence of radiation flux using realistic absorption coefficient and two variants of mean absorption coefficients Kloc P. et al.: Optimization of 3-band Mean Absorption Coefficients  149 emission. This is fairly good agreement for a very low number of bands. In the region of maximum absorption the difference is higher reaching nearly 30%. As a result the amount of radiation escaping plasma is overestimated as is shown in Fig. 5. This can be attributed mostly to the first band which does not capture the ra-diation reabsorption very well. Adding one more band should probably improve the model accuracy. The accuracy of the 3-band MAC approxima-tion can very likely be improved further. The obvious target is the effective radius 𝑟𝑝 in the calculation of Nordborg line limiting factor in (4). Much better agreement could probably be achieved if the effective radius 𝑟𝑝 would be cal-culated by the optimization procedure.  We considered the band boundaries to be tem-perature independent. Additional improvement could likely be made if the band boundaries were calculated for each temperature inde-pendently. This would however significantly increase the complexity of method needed to calculate the radiation transfer. 4 CONCLUSION For the standard definition of Planck mean ab-sorption the position of strong lines is very im-portant and should be considered when decid-ing the band boundaries. Designating one band to a very strong line is often justified. However even with proper band selection the Planck mean average remains fairly inaccurate espe-cially in case of small number of bands. Fairly good accuracy can be achieved with modified Planck mean absorption coefficient and proper band boundary selection for as low as 3-band approximation. The accuracy within 10% in the emitting area can be achieved. The improvement could probably be achieved by including additional band or considering band boundaries being temperature dependent. To improve the accuracy the value of limiting fac-tor or even the MAC inside each band should be calculated by numerical optimization. Acknowledgements  The authors gratefully acknowledge financial sup-port from the Ministry of Education, Youth and Sports under project No. LO1210, project No. CZ.1.07/2.3.00/30.0039 and from Czech Science Foundation under project No. 15-14829S. REFERENCES [1] Randrianandraina H Z, Cressault Y, Gleizes A, J Phys D: Appl Phys 44 (2011) 194012. [2] Nordborg H, Iordanidis A A, J Phys D: Appl Phys 41 (2008) 135205. [3] Bogatyreva N, Bartlova M., Aubrecht V, J Phys: Conference Series 275 (2011) 012009. [4] Jan C, Cressault Y. Gleizes A, Bousoltane K, J Phys D: Appl Phys 47 (2014) 015204. [5] Kloc P, Aubrecht V, Bartlova M, Coufal O, J Phys D: Appl Phys 48 (2015) 055208. [6] Changtong L, Bo Y. J Glob Optim 52 (2012) 45-55. 

Optimization of 3-band Mean Absorption Coefficients

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Optimization of 3-band Mean Absorption Coefficients

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