Coupled radiation and natural convection: Different approaches of the slw model for a non-gray gas mixture

UPCommons Portal del coneixement obert de la UPC http://upcommons.upc.edu/e-prints Aquesta és una còpia de la versió author’s final draft d'un article publicat a la revista Journal of Quantitative Spectroscopy & Radiative Transfer http://hdl.handle.net/2117/367585 Article publicat / Published paper: Colomer, G.; Consul, R.; Oliva, A. Coupled radiation and natural convection: Different approaches of the slw model for a non-gray gas mixture. "Journal of quantitative spectroscopy and radiative transfer", Setembre 2007, vol. 107, núm. 1, p. 30-46. DOI: <https://doi.org/10.1016/j.jqsrt.2006.12.011> © <2022>. Aquesta versió està disponible sota la llicència CC-BYNCND 4.0 http://creativecommons.org/licenses/by-nc-nd/4.0/ Coupled radiation and natural convection: different approaches of the slw model for a non gray gas mixture G. Colomer ∗R. C`onsul, A. Oliva Centre Tecnol`ogic de Transfer`encia de Calor (CTTC) Lab. de Termot`ecnia i Energ`etica, Universitat Polit`ecnica de Catalunya (UPC) ETSEIAT, c/ Colom 11, 08222 Terrassa, Spain [email protected]c.edu — http://www.cttc.upc.edu Abstract The coupling between non gray radiation heat transfer and convection-conduction heat transfer is studied. The Spectral Line Weighted sum of gray gases model (slw) is used to account for non gray radiation properties. The aim of this work is to analyze the influence of the different approaches used when calculating the parameters of the slw model. Such strategies include the use of optimized model coefficients to reduce the number of operations, and the interpolation of the distribution function instead of the use of mathematical correlations. Non gray calculations are also compared to gray solutions using the Planck mean absorption coefficient, which can be also calculated with the slw model. The radiative transfer equation is solved by means of the discrete ordinates method, dom. A natural convection driven cavity is chosen to couple radiation and conduction-convection energy transfer. Several cases, with a significant variation of the ratio between radiation to convection heat transfer, as well as the ratio between radiation to conduction heat transfer, are discussed. Key words: Non gray media, radiation, convection, dom,slw 1 Introduction In the early days of computational fluid dynamics, radiative heat transfer was not taken into account because of the overwhelming amount of compu- ∗Tel: +34 937398192, Fax: +34 937398920, e-mail: [email protected]c.edu Preprint submitted to Elsevier Science 11 October 2006 tational resources it required. The following, natural step, was to consider only radiative exchange between surfaces, which does not involve the solution of the Radiative Transfer Equation (rte). As the computational resources increased, detailed numerical models were conceived, such as the Discrete Ordinates Method (dom), which allowed the resolution of participating media. In the last decade of the last century, feasible, as well as accurate, non gray radiation models, were formulated. These models can be broadly divided in full spectrum models and band models. The former, which include the Weighted Sum of Gray Gases (wsgg) model [1], the Absorption Distribution Function (adf) model [2], the Spectral Line Weighted sum of gray gases (slw) model [3], and the Full Spectrum K-distribution fsk model [4], are accurate enough, particularly in homogeneous cases. The later, which include the Statistical Narrow Band (snb), and the Statistical Narrow Band Correlated−k (snbck) model [5], are more precise than global models, but far more resource demanding. The simplest non gray model ever formulated was the wsgg. Soufiani and Djavdan [6] compared this method to the snb method for combustion applications. While the wsgg predicted well radiative heat source for a hot medium surrounded by black walls, the absorption by a cold gas of radiation emitted by hot walls was generally underestimated. The authors attributed this discrepancy on total absorption to the fact that the weights in wsgg model were always taken at the temperature of the emitting body. Belonging to the full spectrum group, both the slw and adf models are more precise than the wsgg. For instance, the slw weights, depend on the local temperature and could lead to more accurate predictions. Coelho et al. [7] used the slw model to account for turbulence-radiation interaction in a diffusion flame. The inclusion of non gray radiative properties improved the agreement between the computed and the measured data. Soufiani et al. [8] employed the adf model for a mixed laminar convection case, considering an homogeneous gas mixture. Their conclusion was that, by taking radiation into account, the flow is affected significantly. The newest full-spectrum model, up to the author’s knowledge, is the fsk, a rigorous mathematical improvement of the slw method. Essentially, a reordering of the wavenumber integration is performed, and a integration over absorption coefficient (instead of an integration over wavenumber) is performed. This is the same principle on which the slw is based, but in the fsk formulation, highly efficient quadrature methods can be used to carry out the wavenumber integration. In the case of scalable absorption coefficient, this method yields exact results, affected only by numerical errors. The fsk model has approximately the same computational cost of the slw model, once an exhaustive data preprocessing has been completed. 2 On the other hand, band models are more accurate than full spectrum, or global, models. Liu et al. used the snbck model for a methane/air [9] flame. In this flame, which can be considered optically thin, it turns out that radiation model did not have much influence in the outcome. Coelho [10] also used a correlated−kband model, this time with a prescribed temperature and concentration fields, obtained from experimental measures. The works by Liu et al. [9] and Coelho [10], however, pointed out that turbulence and combustion models may have greater influence in the final result than the non gray radiation model for coupled problems. Soufiani et al. [11] also used a correlated−kmodel to solve a forced convection situation, considering pure gases only. In their work, preheating or precooling of the gas in the entrance of a circular duct due to radiation was studied. Comparison with the adf model showed that for the precooling condition the adf performs well, while bigger differences between the band and the global models were obtained for higher temperatures. Among the different non-gray models mentioned above, the slw model stands out for its simplicity of its formulation, its level of accuracy, and for the low (compared to band models) computational resources it requires. The aim of this work is to analyze the performance and implementation of the slw model in detail. When considering a non-gray model, the slw in our case, one realizes that there are plenty of strategies that can be adopted. There are several databases with absorption line data which can be used, one may opt for use fitted distribution functions or calculated ones, one may use high resolution partitions of the absorption coefficient domains or optimized coefficients. . . All of these factors possibly have an impact on the results, which may, or may not, be significant. If radiation is not the only mode of heat transfer, this significance can be different wheter radiation is globally meaningful or not. In order to study the impact of these different approaches of the slw model, this work presents the numerical analysis performed in a thermal driven cavity where the coupling of non-gray radiation and convection is taken into account: first, by varying the convective contribution, and second, by considering two cases, one of which is radiation dominated and the other where radiation and conduction are roughly equally significant. These calculations are compared to gray solutions, to stress the appropriateness of the use of the non-gray slw model, with the different approaches detailed in section 2.3. 2 Mathematical model We consider a non reactive, steady state mixture of gases in order to study the influence of the slw model parameters on a complex problem, which combines also convection and conduction heat transfer. The mixture of gases consists 3 of N2, which is considered transparent to radiation, and CO2and H2O, which absorb and emit, but do not scatter, radiation. The relevant governing equations are adapted from [12], which are valid for reactive flows, and hold for low Mach number, with frictional heating and pressure time variation neglected. 2.1 Convection and conduction transport. For the above mentioned steady-state gas mixture, with average velocity ~u, the mass conservation equation reads ~ ∇·(ρ~u) = 0.(1) Momentum conservation leads to ρ~u ·~ ∇~u =−~ ∇p+ρ~g +~ ∇·~τ (2) where ~τ is the shear stress tensor, and the energy conservation equation reads ~ ∇·(ρcp~uT) = ~ ∇·k~ ∇T−~ ∇·~qr.(3) Thermophysical properties, considered to be temperature dependant, are evaluated by using data compiled in the nasa thermodynamic files. Transport coefficients of molecular fluxes of momentum and heat are calculated assuming mixture-averaged formulation. In order to find the pure species transport properties, the chemkin database [13] is used to generate the coefficients of its temperature expansion. 2.2 Radiation transport. The radiative transfer equation, abbreviated rte (Modest [14], p. 303), is solved in detail, taking into account the wavenumber dependence of the absorption coefficient of CO2and H2O. The monochromatic radiative transfer equation for a non scattering medium is considered. For a given direction defined by coordinates (µ, ξ, √1−µ2−ξ2), the rte in two-dimensional geometries is µ∂Iη ∂x +ξ∂Iη ∂y =−κη(Iη−Ibη).(4) The radiative source term ~ ∇·~qrcan be obtained from the rte solution as ~ ∇·~qr= 4κPσBT4−κGG, (5) 4 where κPIb=RηκηIbη dη and κGG=RηκηGηdη. The monochromatic incident radiation on a control volume is Gη=R4πIηdΩ. This term is then plugged into the energy equation 3. 2.3 Modeling of the absorption coefficient. As stated before, in this work we use the slw model. Following the work by Denison and Webb [3], the monochromatic rte in a purely absorbing medium, equation 4, is considered. The absorption coefficient domain is then divided into mranges, bounded by kj+1 and kjwith kj+1 > kj. A characteristic absorption coefficient value within each range is defined, namely k∗ j= (kjkj+1)1/2, with 1 ≤j≤m. For each absorption coefficient range j, an associated wavenumber range ∆ηjis naturally defined, such that for each η in ∆ηj, the actual absorption coefficient κηlies in the range [kj, kj+1]. Thus, the whole spectrum is divided in mnon overlapping bands, in which the absorption coefficient is assumed to be the constant k∗ j. In order to include the spectral ranges where the gas is transparent, it is assumed that k∗ 0=k0= 0. The absorption coefficient intervals so obtained are referred hereinafter as high resolution partition of the absorption coefficient domain. An integration of equation 4 over all of the wavenumber ranges is carried out, for each absorption coefficient range j. After some manipulation, the rte acquires the form µ∂Ij ∂x +ξ∂Ij ∂y =−k∗ j(Ij−ajIb),(6) where ajis a weight sensible to the particular wavenumber dependence of the absorption coefficient of the media under consideration. The integrated intensity over the whole spectrum, in this model, is simply I=PjIj. Denison and Webb [15] developed the slw method for non homogeneous and non isothermal gases. They use values of k∗ jand ajwhich depend on local conditions. The same methods described by them have been used in this work to obtain the local absorption coefficient and weights. In order to be able to calculate the above mentioned weight aj(and in some cases also k∗ j), and calculate the corresponding local characteristic absorption coefficient k∗ j, it is common to define a so called blackbody distribution function, F, as the integral of the blackbody intensity for all wavenumbers for which the absorption coefficient is below a prescribed value: F(Tb,¯ ξ, k) = 1 Ib(Tb)Zκη(¯ ξ)<k Ibη(Tb)dη. (7) In the above expression, the set ¯ ξcontains all the variables that the absorption 5 coefficient may depend on: temperature, pressure, species concentration. . . The detailed dependence of the absorption coefficient on the wavenumber is needed to compute the function F. Using this distribution function, the coefficients ajare readily calculated as aj=F(kj+1)−F(kj). There are two ways to obtain these distribution functions. First of all, there exist correlations in the literature for the computation of the function Ffor CO2[16,17] and for H2O [18], obtained from a least squares fitting procedure. On the other hand, these distribution functions can be directly calculated from detailed absorption line data. In this work, the latest version of the cdsd1000 [19] and hitemp [20] databases are used to compute the distribution functions for CO2and H2O respectively. These functions have been calculated at several temperatures and concentrations (in the case of H2O), as outlined in a work by Mazumder and Modest [4]. The distribution function at local conditions is then obtained by linear interpolation. 2.3.1 Using optimized coefficients. An alternative way of defining the weights ajand the mean absorption coefficients k∗ jis by means of an optimization procedure [3]. The total emissivity of a gas layer of thickness between 0 and Lis calculated, at a given temperature reference Tref , from the Fdistribution function with a large number of absorption cross sections. Such data is then fitted with, for instance, m0 ranges, with m0significantly lower than m, by means of a non linear least squares procedure, to the function (x)≃ m0 X i=1 aj1−e−k∗ jx,0≤x≤L. (8) These optimized coefficients still can be used for non isothermal, non homogeneous media [7]. In the present work, the total emissivity at a single temperature is considered when finding the optimal ajand k∗ jcoefficients. For an isothermal, homogeneous medium, the weights ajcontain all the information, but for a general situation, also the range boundaries kj+1 and kjare needed. The way to compute the boundaries of the spectral ranges from the weights ajis as follows: first, fit the coefficients for the total emissivity, omitting the transparent band (that is, find ajand k∗ j, 1 ≤j≤m0). Second, compute the transparent band weight, a0= 1 −Paj. Then set k1= 0 and kj=F−1(Pai), for 0 ≤i < j, so, for example, k1=F−1(a0), k2=F−1(a0+a1) and so on. However, it is possible that the fitted mean value k∗ jdoes not lie between kjand kj+1, but this fact apparently does not affect the reliability of the results obtained with this optimization procedure. 6 For the absorption cross section domain partition using optimized coefficients, fitted distribution functions are used: the correlations given by Denison and Webb for CO2[16] and for H2O [18], which are fairly outdated. 2.3.2 Gray absorption coefficient. A wavenumber averaged absorption coefficient for the mixture of the two species can be inferred with the aid of the slw model. This averaged absorption coefficient, κP(Planck mean absorption coefficient) is calculated independently for the participating species CO2and H2O, as κP=1 IbZηκηIbη dη ≃X j{F(kj+1)−F(kj)}k∗ j,(9) which depends also on the temperature. It has been computed using the Fdistribution functions (equation 7) obtained from the latest hitran and hitemp databases [20] for CO2and H2O respectively. The use of the Planck mean absorption coefficient yields faster calculations, although simplifying the non gray behavior of the radiant gas. When the absorption coefficient is modeled as gray, the κPof the mixture is computed as the sum of the individual absorption coefficients, weighted by the molar fraction of each species [4]. 2.3.3 Multicomponent media. Since the absorption coefficient, and therefore the distribution function, are available only for individual species, a model is necessary to deal with multicomponent gas mixtures. Independently of what coefficients aj,k∗ j, for the individual participating species are used, the mixture is treated as a single gas. For such a mixture where CO2and H2O are the only radiating gases, the equivalent coefficients for the slw model are found from the individual coefficients as aij =ai,CO2aj,H2Oand k∗ ij =k∗ i,CO2+k∗ j,H2O[21, 22]. Therefore, the rte to be solved for the mixture is µ∂Iij ∂x +ξ∂Iij ∂y =−k∗ ij(Iij −aijIb),(10) and the spectrally integrated intensity is I=Pi,j Iij, with iand jrunning from 1 to m(or m0if the optimized coefficients approach is used), resulting on m2(or m02)rte equations to be solved. For binary mixtures, if the ratio of the molar fractions of the two species is constant, the convolution approach [21] may be used, reducing the number of times the rte needs to be solved to m. 7 2.4 Dimensionless describing parameters. In this work we characterize the flow by using the time scale concept, which arises from the fact that there are several modes of energy transfer, each one of which occurs at its own time rate. A rough approximation of such scales will help both on characterize the solution and choose an optimal time step when numerically solving the coupled equations 1, 2, and 3 [23]. These time scales are associated with conduction, convection, and radiation, and are τc=ρcpL2 k, τb=sL gβ∆T, τv=ρL2 µ, τr=ρcpL∆T σBT4 0 .(11) The time scales discussed above are related in a very simple way to the more commonly used dimensionless numbers to completely characterize the flow. In the case we have solved, the relevant numbers are the Prandtl number Pr, the Rayleigh number Ra, the Planck number Pl and the temperature ratio φ= ∆T/Tcold. The relation between the dimensionless numbers and the time scales is Ra =τcτv τ2 b Pr =τc τv Pl =τr φτc .(12) 3 Numerical method The discretization of the Navier-Stokes equations is carried out using fully implicit finite volume techniques on cartesian staggered grids. To account for the velocity-pressure coupling, the simplec procedure is adopted. The numerical scheme for interpolation was the simplest, upwind like, in order to facilitate the convergence, which is affected by the high non linearity of the radiative source term ~ ∇·~qr. The mesh was concentrated near the walls, by means of an hyperbolic tangent function. The radiation source term is calculated by solving equation 4, and integrating for all the spectrum by using the slw model. The radiative transfer equation, rte, is solved for a discrete set of directions {µi, ξi}, following the Discrete Ordinates Method (dom). The finite volume technique is used to discretize the spatial part of the rte, and an explicit, step by step procedure, is employed to determine the intensity radiation field. The discretization details can be found in [24]. Orthogonal meshes have been used to solve the rte. The angular quadrature set employed was the Tnby Thurgood et al. [25]. 8 6.3 Radiation vs. conduction heat transfer. The different dependence of the time scales with the cavity length allows us to select a range of lengths in which the ratio between radiation and conduction time scale varies in a large amount. If the length is short enough, conduction time scale will become the fastest one, and consequently the heat transfer will be conduction dominated. The effect of the optical model is plotted in figure 8, for two different lengths. It is clear that, for the L1case, the transparent approximation is preferable than the Planck mean model, while for the bigger, L2cavity, the difference between the models clearly shows up. This is because the optical depth of the mixture is proportional to the cavity size, and thus, the transparent behavior is more evident for the smaller enclosure. In figure 9, the temperature profiles for different cavity sizes, obtained using the high resolution slw method, approach (iv), are plotted, and compared to the profile obtained when radiation is neglected. As we can expect from table 2, for the smallest cavity, in addition that all radiation models tend to the transparent behavior, the results obtained are also close to the purely conduction case, because of the relative significance of radiation and conduction time scales. The L0and L2cavities exhibit a very similar temperature profile, meaning that for cavity lengths larger than L2, radiation transfer dominates over heat conduction. 7 Conclusions Several absorption coefficient approaches using the slw model have been taken into account, in order to consider the effect of non gray radiative heat transfer in a participating medium, on a thermal driven cavity problem. The use of any of the non gray methods is justified since neither the gray gas nor the transparent model captures well the real gas behavior. The optimized coefficients approach (iii) yields results very close to those from the high resolution approach (iv), except for the velocity field for the strongest gravity considered. The use of optimized coefficients is therefore recommended, in order to reduce the number of calculations to perform. Further tests, considering mixed or forced convection, should be done to check the performance of approach (iii) in these cases. In all cases solved, the Planck mean approximation gives the worse results compared to non gray models. This fact can be seen in temperature profile plots, in the velocity profile plots, in the total heat flux table, and in the deviation with respect to approach (iv) table. It has been observed that, increasing the convection contribution (by decreasing the associated time scale τb), the effects of the radiation approach become 15 less significant to determine the temperature field and the total heat flux through the isothermal walls. In these cases, the transparent media approximation seems to offer a good result, both in temperature profiles and total heat flux, and is clearly preferable to the gray gas assumption, albeit being the crudest, cheapest (computationally speaking), model. On the other hand, increasing the gravity strength, the differences on the velocity field with respect to the high resolution slw solution tend to increase. This can be attributed to the fact that the temperature ratio is high (φ= 1), meaning that the flow structure is largely affected by radiation, and also to the fact that the velocity field itself is larger, thus magnifying the differences between the different strategies employed. It has also been checked that, for small enclosures, the choice of the radiation approach has less impact on the outcome, because the participaing gases are nearly transparent. For larger enclosures, heat conduction, whose time scale grows with the square of the cavity size, plays no significant role on the energy transfer. The above mentioned time scales are good parameters to describe what kind of flow is expected, in addition to provide a natural choice of the discrete time step when solving the Navier-Stokes equations. Acknowledgements The research has been financially supported by the ’Ministerio de Ciencia y Tecnolog´ıa’, Spain (ref. TIC 2003-07970). References [1] Michael F. Modest. The Weighted-Sum-of-Gray-Gases Model for Arbitrary Solution Methods in Radiative Transfer. Journal of Heat Transfer, 113:650– 656, August 1991. [2] Laurent Pierrot, Philippe Rivi`ere, Anouar Soufiani, and Jean Taine. A fictiousgas-based absorption distribution function global model for radiative transfer in hot gases. J. Quant. Spectrosc. Radiat. Transfer, 62:609–624, 1999. [3] M. K. Denison and Brent W. Webb. 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Spectral radiative effects and turbulence/radiation interaction in a non-luminous turbulent jet diffusion flame. Combustion and Flame, 133(1–2):75–91, 2003. [8] EzEddine Sediki, Anouar Soufiani, and Mohamed Salah Sifaoui. Combined gas radiation and laminar mixed convection in vertical circular tubes. Int. J. Heat and Fluid Flow, 24(5):736–746, 2003. [9] Fengshan Liu, Hongsheng Guo, and Gregory J. Smallwood. Effects of radiation model on the modeling of a laminar coflow methane/air diffusion flame. Combustion and Flame, 138:136–154, 2004. [10] P. J. Coelho. Detailed numerical simulation of radiative transfer in a nonluminous turbulent jet diffusion flame. Combustion and Flame, 136(4):481– 492, 2004. [11] EzEddine Sediki, Anouar Soufiani, and Mohamed Salah Sifaoui. Spectrally correlated radiation and laminar forced convection in the entrance region of a circular duct. Int. J. Heat and Mass Transfer, 45(25):5069–5081, 2002. [12] K. Claramunt, R. C`onsul, C.D. P´erez-Segarra, and A. Oliva. Multidimensional mathematical modeling and numerical investigation of co-flow partially premixed methane/air laminar flames. Combustion and Flame, 137:444–457, 2004. [13] R.J. Kee, F.M. Rupley, and J.A Miller. The Chemkin Thermodynamic Database. Technical report, Sandia National Laboratories, 1987. [14] Michael F. Modest. Radiative Heat Transfer. McGraw Hill, 1993. [15] M. K. Denison and Brent W. Webb. The spectral line-based weighted-sum-ofgray-gases model in nonisothermal nonhomogeneous media. Journal of Heat Transfer, 117(2):359–365, 1995. [16] M. K. Denison and Brent W. Webb. Development and application of an absorption-line blackbody distribution function for CO2.Int. J. Heat and Mass Transfer, 38(10):1813–1821, 1995. [17] Michael F. Modest and Ranjan S. Mehta. Full spectrum k-distribution correlations for CO2from the CDSD-1000 spectroscopic database. Int. J. Heat and Mass Transfer, 47(10–11):2487–2491, 2004. [18] M. K. Denison and Brent W. Webb. An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer. J. Quant. Spectrosc. Radiat. Transfer, 50(5):499–510, 1993. 17 [19] S. A. Tashkun, V. I. Perevalov, J-L. Teffo, A. D. Bykov, and N. N. Lavrentieva. CDSD-1000, the high-temperature carbon dioxide spectroscopic databank. J. Quant. Spectrosc. Radiat. Transfer, 82:165–196, 2003. [20] L. S. Rothman et al. The HITRAN 2004 molecular spectroscopic databased. J. Quant. Spectrosc. Radiat. Transfer, 96(2):139–204, 2005. [21] M. K. Denison and Brent W. Webb. The spectral line weighted sum of gray gases model for H2O/CO2mixtures. Journal of Heat Transfer, 117(3):788–792, 1995. [22] Laurent Pierrot, Anouar Soufiani, and Jean Taine. Accuracy of narrow-band and global models for radiative transfer in H2O, CO2, and H2O−CO2mixtures at high temperature. J. Quant. Spectrosc. Radiat. Transfer, 62(5):523–548, 1999. [23] Guillem Colomer. Numerical methods for radiative heat transfer. PhD thesis, Universitat Polit`ecnica de Catalunya, 2006. [24] G. Colomer, M. Costa, R. C`onsul, and A. Oliva. Three dimensional numerical simulation of convection and radiation in a differential heated cavity using the discrete ordinates method. International Journal of Heat and Mass Transfer, 47(2):257–269, 2004. [25] C. P. Thurgood, A. Pollard, and H. A. Becker. The Tnquadrature set for the discrete ordinates method. Journal of Heat Transfer, 117:1068–1070, November 1995. [26] Vincent Gouti`ere, Fengshan Liu, and Andr´e Charette. An assessment of real-gas modelling in 2D enclosures. J. Quant. Spectrosc. Radiat. Transfer, 64(3):299– 326, 2000. [27] J. Cadafalch, C.D. P´erez-Segarra, R. C`onsul, and A. Oliva. Verification of finite volume computations on steady state fluid flow and heat transfer. Journal of Fluids Engineering, 124:11–21, 2002. [28] A. Y¨ucel, S. Acharya, and M. L. Williams. Natural convection and radiation in a square enclosure. Numerical Heat Transfer, Part A, 15:261–278, 1989. 18 Fig. 1. Non isothermal enclosure filled with an homogeneous mixture of radiating gases. Wall heat flux (a) and divergence of radiative heat (b) are compared to the results obtained by Gouti`ere et al. [26]. Fig. 2. (a): Results with optimized coefficients, obtained by fitting the total emissivity of a gas layer, compared to normal, high resolution partition of the absorption coefficient domain. (b): Results of interpolated distribution function, obtained from latest absorption line data available, compared to the standard fitted distribution function, using the correlation by Modest and Mehta [17]. Fig. 3. Scheme of the solved thermal driven cavity. Fig. 4. (a): Temperature profile at y= 0.5 m, for all optical models considered. Radiation dominated case, g= 0. (b): Same as left, with g1. Fig. 5. (a): Temperature profile at y= 0.5 m, for all optical models considered, with g5. (b): Same as left, with g9. Fig. 6. (a): Profile of the horizontal component of the velocity at x= 0.5 m, for all the optical models considered, g1case. (b): Same as left, g9case. The difference between the slw and the other models is larger in this situation. Fig. 7. (a): Isotherms (up) and streamlines (down) for the g9case, using the slw approach (iv). (b): Same as left, for the g1case. Fig. 8. (a): Temperature profile at y/L = 0.5, for all the optical models considered. Small cavity L1= 0.025 m. (b): Same as left, for a bigger cavity L2= 0.4 m. Fig. 9. Temperature profile for different size enclosures. Except for the non radiating case, the temperature profiles are obtained with the high resolution slw method. L0= 1 m, L1= 0.025 m, and L2= 0.4 m. 19 CO2H2O k∗ jajk∗ jaj 2.123118e−01 1.557237e−01 2.188490e−01 2.888546e−01 2.615123e+ 00 9.144159e−02 1.952972e+ 00 2.157405e−01 2.803898e+ 01 6.119854e−02 1.378076e+ 01 1.497009e−01 4.652916e+ 02 2.902018e−02 1.273570e+ 02 5.291578e−02 Table 1 Optimized coefficients obtained by fitting total emissivity data to equation 8, with L= 10 and Tref = 600 K. Convection dominated (L0= 1 m) Pl = 0.024, φ= 1, Pr = 0.711 g0g1g5g9 τb/τr∞0.059 0.015 0.0037 Ra/1060 0.686 10.98 175 Conduction and radiation heat transfer Ra = 0, φ= 1, Pr = 0.711 L1L2 τc/τr1.02 16.4 Pl 0.976 0.061 Table 2 Values of the non dimensional relevant parameters for the different configurations. First, the buoyancy time scale and Rayleigh number for the different values of gravity field are shown. In all cases with g6= 0 the convection time scale is the fastest. Last, conduction time scale and Planck number are shown for the different cavity lengths L1= 0.025 m and L2= 0.4 m. 20 g0g1g5g9 Transparent (i) 291.1 294.0 312.0 341.6 Planck mean (ii) 106.5 114.1 163.9 239.7 slw optimized (iii) 325.5 331.2 349.9 377.9 slw (iv) 348.1 354.2 369.9 394.4 Table 3 Non dimensional heat flux through hot wall: the non dimensional factor is the conductive heat flux, k0∆T/L. σ(T)/T0(%) σ(u)/|umax|(%) σ(v)/|vmax|(%) g00.65/2.67/2.57 — — g10.91/3.92/2.53 3.18/13.6/3.38 3.25/14.0/3.55 g50.72/4.39/1.92 5.74/24.8/13.1 5.89/25.8/13.6 g90.50/3.35/1.57 6.81/27.2/18.0 6.22/25.3/16.4 Table 4 Deviation of the different models from the slw solution. Each triplet represents the deviation of the slw with optimized coeffcients, the Planck assumption, and the transparent gas behavior solutions respectively. 21 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 qwall (kW/m2) x(m) Homogeneous, non isothermal H2O-CO2 mixture SLW, 20 gray gases, T7 SLW Goutiere et al. [28] (a) -200 -150 -100 -50 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 div(q) (kW/m3) y(m) Homogeneous, non isothermal H2O-CO2 mixture SLW, 20 gray gases, T7 SLW Goutiere et al. [28] (b) Figure 1 22 -25 -20 -15 -10 -5 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 dq/dx (kW/m3) x (m) Non isothermal, homogeneous 10% CO2-20% H2O mixture SLW optimized 4x4 cross sections SLW double integration, 10x10 cross sections (a) -10 -8 -6 -4 -2 0 2 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 dq/dx (kW/m3) x (m) Non isothermal, homogeneous 10% CO2 medium Interpolated distribution function, 10 cross sections Fitted distribution function, 10 cross sections (b) Figure 2 23 Thot Tcold Adiabatic wall Adiabatic wall g=gnˆ x y Figure 3 24