MODELLING OF RADIATIVE HEAT TRANSFER IN A TURBULENT PILOTED JET DIFFUSION FLAME

Transcription

1 MODELLING OF RADIATIVE HEAT TRANSFER IN A TURBULENT PILOTED JET DIFFUSION FLAME P. J. Coelho Instituto Superior Técnico, Mechanical Engineering Department Av. Rovisco Pais, Lisboa, Portugal coelho@navier.ist.utl.pt O. J. Teerling University of Leeds, Department of Fuel and Energy Leeds, LS2 9JT, U.K. preojt@leeds.ac.uk D. Roekaerts Delft University of Technology, Thermal and Fluids Sciences P.O. Box 5046, 2600 GA Delft, The Netherlands also at Shell Research and Technology Centre, Amsterdam, The Netherlands dirkr@ws.tn.tudelft.nl Corresponding author: P.J. Coelho Instituto Superior Técnico, Mechanical Engineering Department Av. Rovisco Pais, Lisboa, Portugal coelho@navier.ist.utl.pt Phone: (351)

2 Fax: (351)

3 ABSTRACT Radiative heat transfer in a non-luminous piloted turbulent jet diffusion flame was numerically simulated taking the spectral radiative effects and the turbulence radiation interaction into account. It is shown that the optically thin approximation overestimates the radiant fraction by a factor of about 2. The discrete ordinates method along with the assumption of a grey medium yields only a marginal improvement. The spectral nature of gaseous radiation needs to be taken into account to bring the predicted radiant fraction close to the measured one, while the interaction between turbulence and radiation is not very important for the studied flame. Keywords: Radiation/Turbulence Interaction, Turbulent Flames, Discrete Ordinates Method, Gas Radiative Properties

4 NOMENCLATURE a C C ~ Emission weighting factor Absorption cross-section Supplemental absorption cross section C* Non-dimensional radiant power d E F I L L stoich Inner jet diameter Emissive power Absorption-line blackbody distribution function Radiation intensity Path length Stoichiometric flame length N c Molar density of CO 2 N g N w P P ~ q 1, q 2 q R q r R Number of grey gases Molar density of H 2 O Pressure Probability density function Incident wall heat flux Radiative heat flux at the domain boundary Radiative heat flux vector Radial coordinate Radius of the computational domain S rad,exp Measured total radiant power T Temperature

5 w x X X R y Z Quadrature weight Coordinate Molar fraction Local radiative energy loss factor Coordinate Mixture fraction Greek symbols ε η κ κ P ρ σ Ω Emissivity Wave number Absorption coefficient Planck mean absorption coefficient Density Stefan-Boltzmann constant Solid angle Subscripts b Blackbody c CO 2 g loc m ref Gas Local Direction Reference

6 s T w Species Total H 2 O Superscripts m Direction Time-averaged value

7 1. Introduction The numerical simulation of reactive flows often requires a simultaneous coupled fluid flow/radiative heat transfer calculation. The computational and memory requirements of the reactive fluid flow simulation are generally significant, and therefore computationally demanding radiation solvers are not affordable. Accurate radiation models, such as the zone and Monte Carlo methods, may become prohibitively expensive for coupled fluid flow/heat transfer problems. Similarly, accurate gas radiative properties models, such as the statistical narrow band model and the correlated k-distribution method, may be too time consuming for many practical applications. Hence, it is important to find out to what extent less time consuming models can provide the level of accuracy required for engineering calculations. Non-luminous turbulent jet flames have been investigated for a long time, both experimentally and numerically, not only because of their practical relevance, but also because of their simple geometry and configuration. This relative simplicity allows the use of sophisticated turbulence and combustion models, as well as detailed experimental investigations using advanced non-intrusive optical techniques. However, most numerical works have either neglected thermal radiation or used a simple optically thin approximation. In recent years, several workshops on measurement and computation of turbulent nonpremixed flames have taken place [1]. Several test cases were defined and extensively subjected to both experimental and numerical investigation by different research groups. One of these test cases was a piloted turbulent jet diffusion flame. Detailed boundary conditions were obtained and the optically thin approximation, along with the Planck mean absorption coefficient, was recommended to be used for modelling purposes. One of the outcomes of the numerical results, from more than 10 different research groups, was the difficulty to accurately predict the

8 concentration of NO in the so-called flame D [1]. It is well known that the thermal NO formation is strongly dependent on temperature, and therefore an accurate temperature prediction is a prerequisite to accurately predict NO concentration. Since the temperature is influenced by thermal radiation, the adequacy of the optically thin approximation was put into question. Although most people that attended the 4 th Workshop in Darmstadt were convinced that such an approximation was suitable, it was decided to perform additional measurements concerning the radiant fraction, and modellers were asked to predict that fraction. The measured radiant fraction for flame D was 5.1% [2]. Calculations performed using a probability density function (pdf) transport model and the conditional moment closure, along with the optically thin approximation, have predicted values of 10.5% and 12.5%, respectively [2]. However, independent calculations based on a full joint pdf method and the same optically thin approximation gave a value of 2.85% for the radiant fraction [3], accounting only for the region x/d 60, where x is the longitudinal coordinate and d is the diameter of the fuel nozzle. This suggests that the measured value might be approached if the entire flame were considered. Motivated by these contradictory results, we have performed the numerical simulation of this flame using a Reynolds stress secondorder closure, the steady laminar flamelet model and different approaches for radiative transfer [4]. These results have shown that the optically thin approximation strongly overestimates the radiative heat fraction, while the discrete ordinates method (DOM) together with a Planck-mean absorption coefficient performs only marginally better. However, accurate predictions were achieved using the DOM and the spectral line-based weighted-sum-of-grey-gases (SLW) model [5]. Nevertheless, the temperature field and the radiating species, namely CO 2 and H 2 O were overpredicted in the flame region where radiation is more important. Therefore, this might be responsible for the poor performance of the optically thin approximation. In order to clarify this issue, additional calculations

9 were carried out using the measured temperature and species concentrations, and performing radiative calculations decoupled from the reactive fluid flow code. These results are presented here, following the simulation of two preliminary test cases documented in the literature to investigate the accuracy of the SLW model on the simulation of non-grey, non-isothermal and non-homogeneous, multidimensional media. 2. Modelling The model for radiative heat transfer used here combines an advanced radiative properties model (SLW) with an accurate method to solve the time averaged radiative transfer equation (DOM). Special attention is paid to issues related to turbulence-radiation interaction (TRI). An investigation with some similarities with the present one, although based on different models, is described in [6] Gas Radiation Properties Model The radiative properties of the gas mixture are computed using the SLW model. In a mixture with two participating gases, namely CO 2 and H 2 O, the total emissivity, may be written as N g N g k= 0 j= 0 [ 1 exp( κ L) ] ε = a (1) g where indices j and k identify the jth grey gas component for H 2 O and the kth grey gas component for CO 2, N g is the number of grey gases and L is the path length. The values j=0 and k=0 account for the spectral windows where H 2 O and CO 2 are transparent to radiation, respectively. The weight a in Eq. (1) is defined as the fraction of blackbody energy in the spectrum where the effective absorption cross-section of the two species H 2 O or/and CO 2, C w,j and C c,k, has ~ respectively a value in the range ( C w, j ~ ~ C ) and ( C c, k, w, j+ 1 ~, C c, k+ 1 ). It is calculated from the

10 absorption-line blackbody distribution function, which for a species s (w or c), F s, is defined as the fraction of the blackbody energy in the portions of the spectrum where the high-resolution spectral absorption cross-section of the gas, C s,η, is less than a prescribed value C s : F s 1 ( Cs, Tb, Tg, PT, X s ) = Eb (, Tb ) 4 η η η,,, σ T b i ( ) i Cs Tg PT X s dη (2) where σ is the Stefan-Boltzmann constant and E bη is the spectral emissive power of a blackbody evaluated at wave number η and blackbody (source) temperature T b. The subscript i refers to the ith spectral segment where C < s, η Cs, which depends on C s, gas temperature, T g, total pressure, P T, and molar fraction of species s, X s. The correlations for F w and F c given in [7] and [8], respectively, were employed in all the calculations reported below. The joint distribution function of two species is then also known by an assumption of statistical independence. The absorption coefficients κ in Eq. (1) are calculated as κ = (3) N w Cw, j + Nc Cc, k where N w and N c are the molar densities of H 2 O and CO 2, respectively. In the general case of a nonisothermal and/or nonhomogeneous medium, the limiting values ~ C s, j are by construction made dependent on the local conditions via the implicit relation [9] F F s s ~ [ Cs, Tb = Tref, Tg = Tloc, X s = X s, loc, PT = PT, loc] = ~ [ C, T = T, T = T, X = X, P = P ] s, ref b ref g ref s s, ref T T, ref (4) The reference state is determined as the spatial average of the temperature, total pressure, and chemical composition fields. Equation (4) is solved at each spatial location using an iterative method.

11 The total number of gases, N g, ranges between 10 and 20. However, it is possible to achieve good accuracy with only N g =3 by means of an optimization procedure based on the minimization of the squared relative error in emissivity over the range of path lengths typical of the flame under investigation, and for the temperature and species concentrations at the reference state [5]. In such a case, the absorption cross-sections as well as the supplemental absorption cross sections at the reference state are determined from the optimization procedure. Equation (4) is still used to calculate the absorption cross-sections at the local conditions. This strategy was used in the present work. 2.2 Radiation Model The time averaged radiative transfer equation for the jth grey gas component of H 2 O and the kth grey gas component of CO 2 may be written as follows [10]: di ds = κ I + a κ I (5) b where I is the radiation intensity for those grey gas components, s is the direction of propagation of radiation, I b is the blackbody radiation intensity, and the overbar denotes time-averaging. Scattering can be neglected. In the first term, fluctuations in κ and I to a good approximation can be considered uncorrelated, because the fluctuations of these two quantities essentially are determined by spatial correlations over larger distances than the turbulent integral length scale. Then di ds = κ I + a κ I (6) b The absorption coefficient depends on the molar fractions of H 2 O and CO 2, the weight a is a function of the same molar fractions and temperature, and I b is a function of the temperature. Furthermore, the molar fractions of H 2 O and CO 2 and the temperature depend on the local state of the mixture. In the case when the local state only depends on the mixture fraction and an energy loss factor, X R, the required mean values can be computed by means of integration over the mixture

12 fraction range, taking into account all the dependences referred above (X R is a non-dimensional quantity that is defined from local values of enthalpy and species concentrations. It represents the ratio of the local energy released by radiation to the energy that would have been released if the products were cooled down to the room temperature): ( Z ) ( Z, X ) 1 κ ~ κ = ρ P( Z ) dz (7) 0 ρ a R ( Z, X R ) κ ( Z, ) I b ( Z, X R ) ρ ( Z, X ) 1 a ~ κ I = ρ P( Z ) dz (8) b 0 R In the more general case that the local state also depends on other variables, the equations (7) and (8) are still valid if the quantities κ, a and I b inside the integral are the conditional averages at a certain value of mixture fraction. The measured instantaneous values of temperature, H 2 O and CO 2 mass fraction were used to evaluate the integrals in equations (7) and (8). These calculations do not use CFD. Therefore, errors arising from turbulence and combustion models do not influence these radiative calculations, but they rely on an assumed shape of the pdf. We have used a clipped Gaussian pdf shape, defined from the measured mean and variance of mixture fraction. A simpler model for TRI is obtained if κ is calculated using the local mean temperatures and mean species concentrations. The influence of turbulent fluctuations of mixture fraction on radiative transfer 4 4 is still taken into account, because T rather than T is used. However, the TRI is not fully accounted for, because the turbulent fluctuations are ignored in the calculation of the mean absorption coefficient, and the correlation between the absorption coefficient and the blackbody emissive power is neglected. Applying the DOM, the time-averaged equation for the jth component of H 2 O, the kth component of CO 2, and the mth direction can be written in the form:

13 di ds m m = κ I + a κ I (9) b The total radiation intensity for direction m is given by the sum of m I over all j and k. The source term of the transport equation for enthalpy is given by N g N g M m q = 4π a κ Ib κ wm I (10) k= 0 j= 0 m= 1 The volume integral of the source term gives the total radiative heat loss of the flame. 3. Results and discussion 3.1 Validation of the model This section describes the application of the radiation code to two test cases documented in [11] in order to show the performance of the DOM/SLW models. The results calculated using the SLW model are compared with results from [11] obtained using the statistical narrow-band model (SNB) of Malkmus [12], which is considered the most accurate narrow band model [13]. A two-dimensional rectangular enclosure of 1.0 m 0.5 m is considered in both test cases. The walls are black and kept at 0 K. The pressure in the domain is 1 atm. The non-uniform temperature and CO 2 fields (case 1) are given by ( x, y) = 1200 [ ( 1 2 x 0.5 )( 1 4 y 0.25 ) + 1] [ K] ( x, y) = 0.02 [ 4 ( 1 2 x 0.5 )( 1 4 y 0.25 ) + 1] T (11a) X c (11b) A flame-like temperature profile combined with a uniformly distributed mixture of 20% H 2 O and 10% CO 2 is considered in case 2. The temperature profile is defined as T 2 3 ( x, y) = ( x 400)( 1 3 y + 2 y ) [ K] for x 0. 1 o o (12a)

14 T 2 3 ( x, y) = ( x 1)( 1 3 y + 2 y ) [ K] for x > 0. 1 o o (12b) where y o = 0.25-y /0.25. The grid size is the same as in the original reference [11], namely a uniform grid with 61 x 31 cells for case 1 and 81 x 41 cells for case 2. In both cases the T 7 [14] quadrature is adopted. The radiative source terms -( q) x and -( q) y for the non-isothermal, non-homogeneous test case 1 are plotted in Figs. 1a and 1b, respectively. The SLW model with 3 optimized coefficients (Ng=3 in Eq. 1) gives reasonable predictions, except in the vicinity of the walls, where the error is large. However, additional calculations have shown that if 20 grey gases are used instead of 3, then the results near the walls are in good agreement with the reference solution. This suggests that 3 grey gases are not sufficient to satisfactorily reproduce the evolution of the source term near the walls in this test case, even if optimized coefficients are used. The incident wall heat fluxes q 1 (top wall) and q 2 (right wall) for case 1 are plotted in Figs. 1c and 1d, respectively. The predicted heat fluxes at the top wall are within 3% of the benchmark results. At the right wall the SLW model shows a smaller discrepancy (1.2 %) compared with the SNB predictions of the incident heat wall fluxes. These discrepancies of the wall fluxes are measured relatively to the peak values. The evolution of the source terms -( q) x and -( q) y along the centrelines of the rectangular domain studied in test case 2 is shown in Figs. 2a and 2b, respectively. The prediction of -( q) x is relatively good, except for a slight overprediction at the point where the heat source reaches a minimum value. The present model also yields a source term -( q) y very close to the SNB results, except near the walls, where the source term model is slightly overpredicted. The incident wall heat fluxes are shown

15 in Figs. 2c and 2d. The peak values of the incident wall heat fluxes differ by 3.3% and 3.1% from the reference solution at the top and right walls, respectively. These calculations show that for 2D enclosures with non-uniformly distributed temperature and/or species fields the DOM/SLW model (using optimised coefficients) predicts the radiative heat transfer reasonably well, except in the close vicinity of the walls, where a small number of grey gases in the SLW model may be insufficient. In addition to this, it was shown in [11] and [15] that other existing models are either too computationally expensive or far too inaccurate. 3.2 Application to a turbulent piloted jet diffusion flame The turbulent methane/air jet flame considered in this work is flame D, experimentally investigated by Barlow and Frank [16]. The experimental data is available in [1], and the measured mean temperature profiles are shown in Fig. 3. The burner has an inner jet diameter of 7.2 mm and a pilot with an inner diameter of 7.7 mm and an outer diameter of 18.2 mm. The fuel composition is 25% CH 4 and 75% air at 294 K. The annular pilot burns a mixture of C 2 H 2, H 2, air, CO and N 2 with the same enthalpy and equilibrium composition as methane/air at 0.77 equivalence ratio and at 1880 K. There is also a coflow of air. The fuel jet Reynolds number is The flame burns as a diffusion flame, with no evidence of premixed reaction in the fuel-rich methane/air mixture. The mean temperature and species concentration fields are defined from the measured mean temperature, CO 2 and H 2 O mass fractions. The measured mean and variance of mixture fraction allow the specification of the pdf of mixture fraction, provided that a pdf shape involving just two parameters is assumed. A clipped Gaussian pdf shape was assumed in this work. Moreover, the instantaneous data, also available in [1], provides flamelet-like relationships between instantaneous

16 values of temperature/species mass fractions and mixture fraction. This enables the calculation of mean values of I b, κ (Eq. 7) and a κ I b (Eq. 8). The experimental data has been interpolated and extrapolated as appropriate, since it is limited to the centreline profile up to x/d=80 and a few radial profiles up to x/d=75. The interpolation of experimental data has been carried out using cubic splines, which are expected to yield small errors. However, the extrapolation required for distances such that x/d > 75 or 80 is subject to greater uncertainty, which is likely to influence the predictions to some extent. Nevertheless, this approach is expected to introduce much lower uncertainty in the radiative transfer calculations than that introduced by a coupled reactive fluid flow/heat transfer simulation. Figure 4 shows the predicted non-dimensional radiant power C* along the axial direction (L stoich = 47d is the stoichiometric flame length). It is defined as C 2 * 4π R qr = (13) S rad,exp where q R is the radiative heat flux at the domain boundary, which changes along the axial direction, R is the radius of the computational domain, which is equal to the radial distance from the burner axis to the radiometer used in the experiments, and S rad,exp is the measured total radiant power. The predictions for the following modelling approaches, all of them employing the DOM, are shown in Fig. 4: Method 1 Calculations are performed using the Planck-mean absorption coefficient, κ P, calculated as reported in [1], in both the emission and absorption terms, and the TRI is only accounted for via I b. This means that equations (7) and (8) are not employed, and so the influence

17 of turbulent fluctuations on the absorption coefficient is ignored, as well as the correlation between the absorption coefficient and the blackbody emissive power. Method 2 - Calculations are performed using κ P, calculated as reported in [1], and accounting for the TRI as implied from Eq. (6), i.e., using equations (7) and (8). Method 3 Calculations are performed using the SLW model and accounting for the TRI only via I b. Method 4 Calculations are performed using the SLW model and accounting for the TRI as implied from Eq. (6). It is evident from Fig. 4 that there is a major difference between the results computed using κ P (methods 1 and 2) and the SLW model (methods 3 and 4). The first approach strongly overpredicts the radiative heat loss, while the second one is in much better agreement with the experimental data, although it underestimates the measured values. Figure 3 also shows that, for the flame under consideration, the influence of TRI is adequately accounted for via I b, while the influence of turbulent fluctuations on the absorption coefficient is of minor importance, as well as the correlation between the absorption coefficient and the blackbody emissive power. Further insight into the performance of the different approaches is provided by the comparison between the predicted and the measured total radiative heat loss fraction, which is summarized in Table 1. The results given in Table 1 show that the optically thin approximation overestimates the measured fraction of radiative heat loss by a factor of 1.9. However, only a minor improvement is obtained if the DOM is employed together with κ P. In both cases, emission is calculated in the same way, and

18 this implies that the difference between them comes from the absorption term, which is ignored in the optically thin approximation. When the DOM along with κ P is employed, a comparison between the emission and the absorption terms reveals that emission is at least one order of magnitude larger than absorption, except in the coldest regions of the flame. This is the reason why the DOM, which accounts for both emission and absorption, performs only marginally better than the optically thin approximation, as far as the prediction of the radiative heat loss is concerned. The significant difference between the results obtained using κ P and the SLW model, as highlighted in Table 1, needs further explanation. In fact, the emission term is correctly represented by the Planckmean absorption coefficient. However, the absorption term is not. In fact, the use of κ P in the DOM involves the following approximation for the absorption term: κ I dωdη κ P I dωdη (14) η η 0 4π 0 4π However, this approximation may be a crude one, as in the present case. Both terms were calculated for the present flame using the SLW model, and it was found that the first term is about one order of magnitude larger than the second one over a large part of the flame. Hence, it can be concluded that the absorption term is strongly underestimated if it is approximated using a Planck-mean absorption coefficient. This explains why the fraction of radiative heat loss computed using methods 1 and 2, which employ κ P, is much higher than that calculated using methods 3 and 4, which use the SLW model. η Despite of the better results achieved using the DOM/SLW models, the measured fraction of radiative heat loss is underestimated by about 25%. There are several reasons that altogether may explain this discrepancy. These are the uncertainties inherent to the experimental data, enhanced by the need to interpolate and extrapolate that data over the whole domain; the assumed pdf shape for

19 the evaluation of the integrals in equations (7) and (8); the modelling assumptions of the SLW model; and the radiation from CO and CH 4, which has been neglected in our analysis. 4. Conclusions Radiative heat transfer in a non-luminous turbulent piloted jet diffusion flame was numerically simulated using several different modelling approaches. The temperature and the species concentration fields were prescribed based on experimental data, in order to prevent uncertainties arising from these fields to influence the conclusions related to the performance of the different radiation methods. It was found that the optically thin approximation overestimates the fraction of radiative heat loss by a factor of 1.9. The discrete ordinates method, together with a Planck-mean absorption coefficient, yields only a marginal improvement of the predictions. The spectral line-based weighted-sum-of-grey-gases yields better results, owing to a more accurate calculation of absorption, but still underpredicts the measured radiative fraction by about 25%. Possible reasons for this discrepancy were given. The interaction between turbulence and radiation was also investigated, and it was shown that it is sufficient to take into account the influence of turbulence fluctuations on the blackbody radiation intensity. 5. Acknowledgements Part of this work was financially supported by the European Commission in the framework of th TMR Network Contract No. ERBFMRX-CT (RADIARE), and by the PRAXIS XXI Programme of the Portuguese Ministry of Science and Technology under contract PRAXIS/EME /12034/1998.

20 REFERENCES [1] [2] Frank JH, Barlow RS, Lundquist C. Radiation and nitric oxide formation in turbulent nonpremixed jet flames. Proceedings of the Combustion Institute 2000;28: [3] Tang Q, Xu J, Pope SB. Probability density function calculations of local extinction an NO production in piloted-jet turbulent methane/air flames. Proceedings of the Combustion Institute 2000;28: [4] Coelho PJ, Teerling OJ, Roekaerts D. Spectral radiative effects and turbulence/radiation interaction in a non-luminous turbulent jet diffusion flame. Combustion and Flame 2003;133: [5] Denison MK, Webb BW. A spectral line-based weighted-sum-of-gray-gases model for arbitrary RTE solvers. J. Heat Transfer 1993;115: [6] Mazumder S, Modest MF. A probability density function approach to modelling turbulenceradiation interactions in nonlumimous flames. Int. J. Heat and Mass Transfer 1999;42: [7] Denison MK, Webb BW. An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer. J Quant Spectrosc Radiat Transfer 1993;50: [8] Denison MK, Webb BW. Development and application of an absorption-line blackbody distribution function for CO 2. Int. J. Heat and Mass Transfer, 1995;38:

21 [9] Denison MK, Webb BW. The spectral line-based weighted-sum-of-gray-gases model in nonisothermal nonhomogeneous media. J. Heat Transfer 1995;117: [10] Modest MF. Radiative Heat Transfer, New York: McGraw-Hill, [11] Goutiere V, Liu F, Charette A. An assessment of real gas modelling in 2-D enclosures, J Quant Spectrosc Radiat Transfer 2000;64: [12] Malkmus W. Random band Lorentz with exponential-tailed S -1 line intensity distribution function. J. Optical Society of America 1967;57: [13] Soufiani A, Hartmann J-M, Taine J. Validity of band model calculations for CO 2 and H 2 O applied to radiative properties and conductive-radiative transfer, J Quant Spectrosc Radiat Transfer 1985;3: [14] Thurgood CP. A critical evaluation of the discrete ordinates method using HEART and TN quadrature, Ph.D. Thesis, Queen s University, Department of Chemical Engineering, Kingston, U.S.A, [15] Coelho PJ, Teerling J, Ströhle J, Quintino T. A comparative study of non-grey gas radiation models, 6 th Int. Conf. Technologies and Combustion for a Clean Environment, 9-12 July, vol. II, 2001, p [16] Barlow RS, Frank JH. Effects of turbulence on species mass fractions in methane/air jet flames, Proceedings of the Combustion Institute, 1998;27:

22 FIGURE CAPTIONS Figure 1 - Radiative source terms along the planes y = 0.25 m and x = 0.5 m of the rectangular domain, and incident heat fluxes along the top (q 1 ) and (q 2 ) right walls for test case 1 (solid line: reference solution [11]; dashed line: present calculations). Figure 2 - Radiative source terms along the planes y = 0.25 m and x = 0.5 m of the rectangular domain, and incident heat fluxes along the top (q1) and (q2) right walls for test case 2 (solid line: reference solution [11]; dashed line: present calculations). Figure 3 - Measured axial and radial temperature profiles [1, 16]. Figure 4 - Predicted and measured non-dimensional radiant power along the axial direction (solid lines: predictions; symbols: experimental data)

23 Fraction of radiative heat loss Measured data Optically thin approximation Method 1 Method 2 Method 3 Method 4 5.1% 9.5 % 8.7 % 8.0 % 3.8 % 3.9 % Table 1 Predicted and measured fraction of radiative heat loss.

24 a) b). (W/m ) x 3 -( q) -5.0x x x10 5 SNB SLW (y = 0.25 m) -2.0x x (m) (W/m ) 3 y -( q). -5.0x x x10 5 (x = 0.5 m) -2.0x y (m) 1.2x10 4 c) 9.0x10 3 d) 1.0x x10 3 q (W/m ) x10 3 q (W/m ) x x x x x (m) y (m) Figure 1

25 2.5x a) 5.0x b) (W/m ) 3 x -( q) x x10 (W/m ) 3 y -( q). -5.0x x x x10 5 (y = 0.25 m) -2.0x10 5 (x = 0.5 m) -1.0x x (m) -2.5x y (m) 2.5x x10 4 c) d) 1.2x10 4 q (W/m ) x10 4 q (W/m ) x x x x x (m) y (m) Figure 2

26 2000 a) 2000 b) T (K) 1000 T (K) 1000 x/d = 60 x/d = x/d x/d = 15 x/d = 30 x/d = r/d Figure 3

27 C* x/l stoich Figure 4