Modelling radiative heat transfer in thermal plasmas Jean-Gaël Lacombe 1, Yves Delannoy 1, Christian Trassy 1 1 IPG-CRS, SIMAP-EPM, Rue de la piscine 38402 Saint Martin d Hères, France jean-gael.lacombe@hmg.inpg.fr The aim of our study is to model radiative heat transfer in pure argon thermal plasmas. A spectral model has been developed for argon lines which undergo absorption. Emission and absorption profiles for argon lines are thus examined. Radiative processes for which the plasma is transparent are taken into account with a et Emission Coefficient (EC) whereas the others are treated with a diffusive spectral model, well adapted to optically thick medias. The relative contributions of the different radiative phenomena are presented in several plasma torch configurations. An experimental setup is presented in order to check the validity of the models used. The measures carried out are compared to spectrum calculations thanks to post treatment tools, including Abel s inversion. 1. Introduction The calculation of the radiative heat transfer in thermal plasma by direct resolution of the radiative transfer equation (RTE) requires prohibitive calculation time (see e.g. [1]). Thus, simplified models have been developed. Among them is the et Emission Coefficient (EC) method which treats radiation as an energy sink assuming photons escape the plasma along an isothermal path [2]. This approach is efficient when absorption is weak. The P1 spectral model, described by [3] for isothermal plasmas can describe the absorption phenomena assuming radiation is isotropic. We propose here an extension of the P1 spectral model to anisothermal plasmas. The radiative heat transfer in pure argon is studied for two plasma torches used for chemical analysis and for silicium purification, which mainly differ by their size and input power. In both case, the P1 model is used for strongly absorbed radiative transitions and the EC is applied to transitions for which the plasma is optically thin. Local thermodynamic equilibrium (LTE) is assumed. As a consequence of this hypothesis, the Kirchhoff s law is used to link emission and absorption and a Boltzmann population of the argon excited states is assumed. In order to determine whether the plasma is optically thick or thin for a given radiative transition or a given wavelength, the argon spectrum has been calculated, including continuum [4], and line broadening phenomena are taken into account [5]. Optical emission spectroscopy (OES) measures have been carried out, in order to check the argon spectrum calculations. A code is used to post treat and reproduce those measures and to determine the effect of the experimental setup on the results [5]. 2. Spectrum calculation To calculate the argon spectrum, free-bound, free-free and bound-bound transitions are taken into account. According to [4], continuum radiation is mainly due photoionisation and Brehmstrahlung. Both phenomena have been calculated between 100 and 1000 nm using the method proposed by [6], including Gaunt and Biberman corrective factors [6,7]. The calculated argon continuous spectrum at 10000K is shown on figure 1. Strong variations of this coefficient around 100, 450 and 700nm are due to radiative recombination [6]. For line radiation, Doppler, Van der Waals and Stark broadening effects have been found to be significant. Stark effect is described using the theory proposed by [8] when the Stark parameters are available. The effect of ions is thus represented by the quasi-static theory, while the effect of electrons is calculated with the impact theory. Stark parameters are not known for all argon lines, but the Stark effect is very significant (figure 2). Thus an Figure 1: argon continuum spectral absorption coefficient at 10000K. approximate correlation is used when Stark
parameters could not be found. This correlation has been presented by [5]. Van der Waals effect is taken into account with the Lindholm-Foley impact theory. The relative contribution of those phenomena is represented on figure 2 for the argon lines that were found to be absorbed in an analysis torch (2cm diameter). Stark parameters could not be found for half of those lines. It appears that argon lines with a lower level at 0eV are much less broadened than the other ones. Once the argon absorption spectrum is obtained, and under the LTE assumption, Kirchhoff s law can be used to link absorption and emission [9]. Figure 2: Half Widths at Half Maximum (HWHM) at 10000K sorted by phenomena for absorbed argon lines in a 1cm ICP torch. 3. Radiation Transport Modelling 3.1 Optically thin approximation The EC has been used to describe the energy loss by radiation when absorption is negligible. The criterion for low absorption is based on optical depth at a given wavelength: τ = κ ds < 0.2 (1) Where τ is the optical depth, κ the spectral absorption coefficient and ds the optical path in the plasma. The criterion depends on the torch geometry but in our case study, most of the argon lines don t undergo absorption. 26 argon lines appeared to be absorbed in analysis torches (with 2cm diameter) and 60 lines in larger process torches (with 4cm diameter). One should remark that the absorbed lines are the most energetic ones. Thus, they may not be negligible in the global radiative heat transfer. The optical depth for continuum radiation is in both cases close to 10-2 and its absorption will not be taken into account. To calculate the energy sink 4πε ( T) due to non absorbed radiative transitions, the classical formula for EC has been used [2]: κ R p 4πε ( T ) = 4π ε e d (2) non absobed lines 0 (2) may be simplified with the assumption of small optical depth (1) into: 4πε ( T ) = 4π ε d (3) non absobed lines 0 In (2) Rp is the plasma radius, which is difficult to define. The plasma is supposed to be isothermal along this radius. Applying the EC method to transitions for which the attenuation factor (the exponential term in (2)) is close to unity makes the assumptions above unnecessary. The EC has been calculated separately for lines and continuum and both were used in the calculations. Figure 3 shows the energy sink as a function of temperature obtained with the EC for continuum and absorbed lines and with the P1 spectral model in a 850W analysis torch. The total energy sink is compared to the results presented in [2] with the EC method for all transitions in a 15mm pure argon plasma. Our model seems to be in good agreement with previous works. The P1 energy sink has an irregular shape because the radiative balance depends on the temperature and on Figure 3: EC as a function of temperature for continuum and non absorbed lines in two different torch geometries the incident radiation rather than on the sole temperature. This comes from the fact that absorption is taken into account as we will show in the next part of this paper. 3.2 Absorbed transitions For the argon lines that are absorbed in plasma, the P1 spectral model is used to calculate heat transfer. Radiation is then supposed to be isotropic, which is true for emission but questionable for absorption. The P1 model is based on the decomposition of spectral radiative intensity into spherical harmonics in order to give a diffusive form
to the RTE [3]. In an axisymetric configuration, this equation takes the following form: G 1 G ( ) + ( ) = κ (4πB G ) (4) z 3κ z r r 3κ r Where G is the spectral radiative intensity integrated over the whole solid angle and B the Planck function. The energy sink term on the right side of the equation in depends on the temperature field. This explains the fact that the total energy sink shows variations around the energy sink calculated with the EC method (figure 3). As absorption is relatively low in pure argon (figure 5), those variations are limited. However, in mixtures with more radiative components such as oxygen, those variations are expected to be more important. Spectral discretization is still prohibitive at this point but [3] proposes to reduces the number of spectral intervals in an isothermal plasma by grouping all parts of the spectrum with the same spectral absorption coefficient. This implies that the resulting grouped spectral intervals have the same spectral absorption coefficient. Then B is summed over all the grouped intervals. It is possible to extend this grouping method to anisothermal plasmas if one can find parts of the spectrum with same absorption coefficient at any temperature. This leads to 3 criterions: - absorption coefficient at the central wavelength must be equal at a given temperature - the upper and lower levels of the transitions must be equally populated - line broadenings must be equal (Doppler and collisional broadenings). and 9 equivalent lines in large torches (instead of 26 and 60 lines respectively). The error margin in applying the criterion is 10%. Each equivalent line is treated with several spectral intervals. In the end the P1 equation needs to be solved on 30 to 50 grouped spectral intervals depending on the precision of the discretization. Gas inlet Figure 5: plasma zones where absorption exceeds emission (W/m3) for 1410W input power in a 2cm diameter analysis torch, radiative losses are concentrated beyond the line absorption = emission Simulations using the models described above have been carried out using the commercial software Fluent, to which an induction module has been added [10]. The plasma simulations presented below use most of the features described in [11]. Figure 4 shows the temperature fields obtained with the P1 spectral method and by applying the EC to all transitions. In both cases, the temperature fields are in good agreement with what could be found in literature [12]. Maximum temperatures, obtained in the coupling zone, are close to 11000K in both cases. Gas inlet Figure 4: temperature field in a 40 mm torch with 12kW input power without absorption (below) and with the P1 spectral model (top) Applying those criterions to the absorbed lines in argon made it possible to reduce spectral discretization to 7 equivalent lines in small torches Figure 6: relative contributions of the different radiative phenomena for different torch geometries and input power. Absorption is relatively week in pure argon, but is expected to be much more important in Ar-O 2 mixtures which the goal of our studies. In zones of the plasma presenting a very strong temperature
gradient, absorption overcomes emission as one can see one figure 5. The results of our calculations are shown on figure 6. It sums up the radiative losses due the different transition types in different torch configurations. It appears that continuum and lines are significant in the thermal balance of the plasma. The radiative losses computed using EC for all transitions come from a EC database provided by the Sherbrooke Department of Chemical Engineering. Our calculations show that the 26 (60) absorbed lines in a 2cm (4cm) diameter plasma are responsible for higher loss than the several hundreds of non absorbed argon lines, although part of their energy is trapped in the plasma. Continuum radiative losses appear to be significant as well. 4. Experimental setup 4.1 Optical emission spectroscopy (OES) In order to control our spectrum calculations, experiments have been carried out. The first step of our verifications is to compare the calculated and measured line broadening. As the measures made with OES are spectral intensities integrated over a segment of the plasma, data has to be treated to obtain the line broadening at a given point. One may use Abel inversion to do so. This inversion consists in a derivation of the observed intensity profiles. Thus, it is very sensitive to small variations of the profile and the data has to be smoothed before the inversion. The resulting profiles for the Ar 451nm argon line at different radial positions after smoothing and inversion are shown on figure 7. This line was chosen because it is intense but not enough to be absorbed, so that one can gather information on the central part of the plasma. The light emitted at the centre of plasma by an absorbed line is more difficult to observe with OES. To compare measured and calculated spectrum, it is necessary to determine the temperature at which the lines were observed. Temperature measurements in pure argon plasma often lead to great uncertainties. According to calculations using the spectral P1 model, the temperature field in the fireball of our experimental plasma lays between 8000K and 9000K, which gives a starting point for our comparisons. Calculations also have a large uncertainty but the temperature field in the fireball is not very sensitive to the calculation parameters (but its volume is). According to those calculations, the Ar 451nm line is expected to be 80 må broad at 9000K. The measured profiles show a larger broadening than the simulated ones by a factor 2. Which is probably due to the effect of the spectrometer slits, the effect of which is not yet taken into account. Order of magnitude calculations show that instrumental broadening may be as high as 80mÅ. Thus the next steps of our work will be to calculate precisely this effect and estimate the real accuracy of our spectrum calculations. 5. Conclusion The P1 spectral model makes it possible to study the different radiative phenomena in plasmas without the help of arbitrary parameters like the plasma radius in the EC method. The latter is however useful as soon as the plasma is transparent for a given transition. The radiative losses in analysis torches are about 10% of the input power but increase strongly with the torch diameter to reach about half of the input power in process torches. The experimental setup shows the spectrum calculations are in the right order of magnitude but needs to be improved for quantitative comparisons. This is the next step of our study. Measuring the intensity on an absolute scale for lines and continuum should also be done in order to check the computed spectral absorption coefficients and to evaluate to global radiative losses of the plasma by an experimental means. 6. References [1] J. Menart, J.Q.S.R.T 67 (2000) 273-291 Figure 7: Ar 451nm emission profiles at different radial positions (center of the plasma R = 0cm) The measured broadenings are between 180 må and 220 må. [2] A. Essoltani, P. Proulx, M.I. Boulos, A.Gleizes, Plasma Chem Plasma Process 14 (3) (1994) 301-315 [3] M.F. Modest, Radiative Heat Transfer (ed. Elsevier Science) (2003) pp 465-494. Academic Press, San Diego.
[4] Cressault, PhD, Paul Sabatier University (Toulouse, France) (2001) [5] C.Trassy, A. Tazeem, Spectrochim. Acta, Part B 54 (1999), 581-602 [6] D. Hofsaess, J.Q.S.R.T. 19 (1978) 339-352 [7] D. Schluter, Z. Phys. D Atoms, Moelcules and Clusters 6 (1987) 249-254 [8] H.R. Griem, Spectral line broadening by plasmas (ed. Academic Press) (1974) pp 320-364. Academic Press, ew York. [9] M.I. Boulos, P. Fauchais, E. Pfender, Thermal Plasmas Fundamentals and Applications (ed. Plenum Press) (1994) Volume 1 pp 325-381. Plenum Press, ew York. [10] Y. Delannoy, C. Alemany, K.I. Li, P.Proulx, C.Trassy, Solar Energy Materials and Solar Cells 72/1-4 (2002) pp.69-75 [11] D.Pelletier, Y.Delannoy, P.Proulx, 4th International Conference on Electromagnetic Processing of Materials, Lyon, France, (14-17 October 2003) 107-112. Publ. Forum Editions, Paris, Fr CODE: 69FIF. [12] J.Mostaghimi, P. Proulx, M.I. Boulos, umerical Heat Transfer. 8 (1985) 187-201