Pressure-dependent weighted-sum-of-gray-gases models for heterogeneous CO2-H2O mixtures at sub-and super-atmospheric pressure

The effect of total pressure of gas mixture is included in the development of new coeﬃcients for the weighted-sum-of-gray-gases model (WSGG). The WSGG formulation previously reported by Bord-bar et al. (combustion and Flame 2014, V. 161, pp. 2435–2445), which accounts for variations of molar fraction ratio of H 2 O to CO 2 , was employed here to obtain a new total pressure-dependent WSGG model. Hence, the new model includes both the effect of total pressure and variation of molar fraction ratio. High-resolution absorption spectra of gases produced by line-by-line (LBL) calculations using the HITEMP2010 spectral database are used to produce the total emissivity databases needed for the WSGG model development and also to produce the benchmark solution of one-dimensional slab problems used for validation of the new model. The performance of the new WSGG model is studied through several test cases representing various conditions of total pressure, inhomogeneity of temperature, concentration of gas species and molar fraction ratios. In all cases, the new model exhibits a good agreement with the LBL solutions. The new WSGG coeﬃcients allow the model to eﬃciently solve the spectral thermal radiation in both sub-and super-atmospheric combustion systems.


Introduction
With recent increment in computational resources, the role of numerical modeling in design process of energy conversion systems has more and more highlighted [1,2] .With an increasing trend during the last three decades, numerical modeling exhibited promising capability to analyze various physical phenomena occurring in energy conversion systems including the pollutants emission [3] , multi-phase flow [4][5][6] , turbulence [7,8] and thermal radiation [9][10][11] .
Thermal radiation is the major heat transfer mode in combustion processes, due to the high temperatures involved and the presence of species that emit and absorb radiative energy, such as carbon dioxide, water vapor and soot [2,3,12,13] .The determination of the radiation field entails the solution of the radiative transfer equation (RTE), which is an integro-differential equation in space and in two directional coordinates with local origin [14,15] .More-over, the radiative properties (in particular, the absorption coefficient) of diatomic and polyatomic molecules present a complex dependence on the radiation spectrum, often characterized by hundred of thousands to millions of spectral lines, whose positions and intensity are in turn dependent on the local thermodynamic state [10] .To obtain the total radiation intensity and related quantities (e.g., the radiative heat flux and the volumetric radiative heat source), the RTE needs to be additionally integrated over the entire spectrum, which, for species as CO 2 and H 2 O , introduces another formidable level of difficulty to the problem.
While radiation modeling of air-fuel combustion at atmospheric conditions is already a challenging task, it can be further complicated for oxy-fuel combustion and for high-pressure conditions.In the former, the possibility of large variations in the chemical concentration of the participating species and its effect on the radiation absorption spectrum need to be accounted for.This can be particularly significant for species such as H 2 O , whose absorption spectrum presents a non-negligible self-broadening effect [16] .The total pressure also affects, even more substantially than changes in mole fraction, the absorption spectra of all participating species, by increasing the values of the absorption coefficient and smoothing its highly-irregular spectral variation [17,18] .
The aforementioned complex spectral dependence of the absorption coefficient of some species can be captured with a high degree of accuracy in the integration of the spectral radiation intensity by the line-by-line (LBL) methodology.In the LBL calculations, detailed information of the absorption lines is extracted from high-resolution spectral databases, so as to allow each one of them to be individually considered.However, the need of solving the RTE many thousands of times for each optical path makes the computational cost of the LBL methodology prohibitive for most applications [11] .This justifies the rise and broad use of models to represent the wavenumber dependence of the radiative transfer is an approximate and less computationally-intensive manner.
Recent years have seen much effort invested in developing or adapting existing spectral models for applications outside of airfuel, and atmospheric combustion.Narrow-band models-and, particularly, the statistical narrow-band (SNB) model [19,20] -have been successfully applied for oxy-fuel combustion at high-pressure conditions (as in, e.g., [21] ), with results showing errors of about 2% relative to the LBL solution for total pressures as high as p = 30 atm .The SNB correlated-k distribution (SNBCK) model, which is based on a reordering the spectral absorption coefficient of each narrow-band into a monotonically increasing function in order to simplify and reduce the computational cost of the traditional SNB model, was also found to be accurate for both oxy-fuel and highpressure combustion.For instance, Kez et al. [22] carried out the first analysis of the SNBCK model for three-dimensional cases representative of high-pressure (20 atm ), oxy-fuel combustion problems; using the SNB model as the reference, the errors of the SNBCK solution were of at most 2.5%.Additional tests of the SNBCK model in similar conditions are reported in [21,23,24] .
Wang and Modest [25] developed a new narrow-band kdistribution database for CO 2 and H 2 O for total pressures ranging from 0.1 to 30 bar , based on which narrow-band and full-spectrum k -distributions can be obtained for non-homogeneous mixtures.Using this database, the full-spectrum k -distribution (FSK) model was applied [26] to predict the radiative heat source in twodimensional, homogeneous CO 2 -H 2 O mixtures at a total pressure of 30 atm , with maximum errors relative to the LBL solution of 11%.Although the medium composition in [26] was inline with characteristic of air-fuel combustion, subsequent studies showed that the FSK model is capable of providing accurate solutions for both atmospheric [27][28][29] and high-pressure oxy-fuel combustion scenarios [21][22][23][24] .Following the work of [25] , the narrow-band k -distribution database was updated in [30] by employing more complete spectroscopic data and expanding it to a wider range of pressures.
The absorption-line blackbody distribution function (ALBDF), necessary for the application of the spectral line-based weightedsum-of-gray-gases (SLW) model [31,32] , has also recently been updated for the full mole fraction range for CO 2 , H 2 O and CO for total pressures between 0.1 atm and 50 atm [33,34] .Chu et al. [23] also applied the SLW model to a set of high-pressure, oxy-fuel problems, finding an accuracy comparable to the FSK model as long as a sufficient number of gray gases is used.Furthermore, because [23] did not use the most up-to-date ALBDFs of [33,34] , it is possible that the performance of the SLW is even better.
In contrast to all these models, the weighted-sum-of-gray-gases (WSGG) model has yet to be consolidated for the treatment of high-pressure conditions.This model is based on a similar principle as the FSK and SLW models.However, because it presents a simpler formulation and smaller computational data and costs, it likely is the most widely used global spectral model for combustion applications.The literature reports various well-established WSGG correlations for mixtures representative of air-fuel and oxy-fuel combustion at atmospheric pressure (see, e.g., [35][36][37][38][39][40] ), with many studies attesting their accuracy at these conditions [41][42][43][44] .However, as it was shown in [21][22][23] , using WSGG correlations developed for atmospheric pressure for high-pressure applications leads to considerable errors.
Hence, attention toward development of WSGG formulations for high-pressure applications has increased in the last decade, and the main studies on the subject are summarized in Table 1 .The pioneering work on this field was the one by Bahador and Sunden [45] , who developed WSGG correlations for pure H 2 O and CO 2 -H 2 O mixtures at total pressures ranging between 1 atm and 20 atm .The correlations were generated using an approach that is typical for WSGG model, by fitting total emittance to a reference data, which in [45] were extracted from the HITEMP1995 [46] and CDSD-10 0 0 [47] spectroscopic databases.A similar effort, albeit using the more accurate HITEMP2010 [48] database for the emittance fittings, was undertaken by Coelho and França for pure species [49] and for CO 2 -H 2 O mixtures characteristic of methane-air combustion [50] .Furthermore, differently from [45] , which only presented comparisons of the total emittance, radiative transfer calculations of a set of one-dimensional test cases were carried out in [49,50] , with their model showing a good level of agreement to the LBL method.
Shan et al. [51] developed the first WSGG formulation for highpressure, oxy-fuel combustion scenarios, based on emittance fittings to the EM2C narrow-band model.Following the same approach as some WSGG formulations for atmospheric pressures [36,52] , these fittings were constructed for fixed H 2 O / CO 2 mole ratios M r and a stepwise interpolation methodology was devised for dealing with varying-M r scenarios.Instead of generating WSGG coefficients for discrete total pressure values as the aforementioned papers did, Shan et al. [51] let the coefficients be dependent on total pressure itself, within a range 1 bar ≤ p ≤ 30 bar .A similar model was introduced by Wang and Xuan [35] for aircraft engine applications, considering the same total pressure range, but only for a single M r .Both [51] and [35] report good accuracy of their respective models in one-dimensional radiative transfer calculations.
The present study proposes a new WSGG formulation for CO 2 -H 2 O mixtures representative of high-pressure combustion.This is a direct extension of the WSGG correlation for atmospheric pressure previously reported by Bordbar et al. [40] to other values of total pressure ( p), and, as depicted in Table 1 , it covers a wider range of total pressures, moleratios and temperatures than the WSGG formulation of Shan et al. [51] .Moreover, the new model is based on the HITEMP2010 [48] spectral database, which is more accurate than the EM2C narrow-band model used as a reference in that paper.Also, by incorporating the M r dependence into the WSGG coefficients (rather than generating coefficients for discrete moleratio values, as in [51] ), it provides a more convenient treatment of problems where M r varies throughout the domain.
Besides high-pressures, the pressure range of the formulation developed here also includes sub-atmospheric conditions (as low as 0.1 atm ).While radiative transfer calculations at these conditions are rare outside of meteorological applications [14] , subatmospheric pressures have often been used as a surrogate to study microgravity combustion (for example, [53][54][55] ), so an accurate treatment of radiation at these low pressures may prove to be important.However, spectral modeling at sub-atmospheric pressures is still a fairly unexplored subject of study, and, as far as the authors' know, this is the first WSGG model that contemplates these conditions.

Model development
To obtain the emissivity database needed for development of new pressure-dependent WSGG model, the line-by-line integration method is implemented to produce the required absorption spec- tra of the gases by using the HITEMP2010 spectral database [48] .
The details of the line-by-line calculation of the absorption spectra of gas mixtures is well described in the literature (see, for instance, [10,39,40,50] ).However, for the sake of completeness, this section first briefly reviews the line-by-line calculations and its characteristics in the present research, and then explains how they are used for development of the new WSGG model.

Determination of the line-by-line based emissivity database
The spectral absorption coefficient of CO 2 and H 2 O in each spectral segment depends on the wavenumber, temperature, the partial pressure of the gases and the total pressure of the gas mixture.The properties of a spectral line i of CO 2 and H 2 O including the line intensity S i , the line center wavenumber ν i , the Lorentz broadening half-width γ L , and the Doppler broadening half-width γ D , can all be obtained or calculated from spectral databases, i.e.HITEMP2010 [48] .In the present work, depending on the relative importance of Doppler broadening to Lorentz broadening, two different line profiles, Lorentz and Voigt, have been implemented in the line by line calculation of absorption spectra of combustion gases.The theory of calculating spectral absorption coefficient along the lines by these two line profiles are described below.
Assuming the combined effect of natural line broadening and collision broadening, the Lorentz line profile evaluates the absorption coefficient at an arbitrary wavenumber ν from a particular spectral line with line center at ν i as where κ ν,i is the absorption coefficient for the i th spectral line at the wavenumber location ν, S i is the molecule-based intensity of the i th spectral line, N(p, T ) is the number density, γ L,i is the Lorentz broadening half width of this spectral line, and ν i is the spectral line location.Applying the ideal gas assumption, the number density is calculated as RT p Y g where N A , R, Y g , p and T represent the Avogadro's number, the gas constant, the gas mole fraction, pressure and temperature, respectively.The temperaturedependent intensity of each spectral Lorentz line is given as where Q is internal partition function of the molecule at temperatures T and T re f , E ν is the lower-state energy of the line, c is the speed of light in vacuum, and k and h are the Boltzmann and Planck constants, respectively.A reference temperature of T re f = 296 K is set for the present LBL calculations.The collision broadening of the spectral line depends on temperature, partial pressure and total pressure as where γ air and γ sel f are the line air-and self-broadening, respectively, n is the temperature-dependence coefficient, and p s is the partial pressure of the species.
The spectral-line profile to produce the histograms for each line is defined based on the relative importance of the Lorentz to Doppler line-broadening effects.Using narrow-band transmissivities as a comparion criterion, Wang and Modest [17] reported that at low pressures ( < 1 bar ) and high temperatures ( > 2500 K), the Doppler broadening may become an important mechanism and therefore the combined Lorentz-Doppler line profile (or Voigt) should be used.In contrast to the Lorentz profile, the Voigt profile does not have a closed form and is instead given as The Humleek algorithm [56] as given in [14] is used to calculate the Voigt profile in the present work whenever needed.Because, in general, the Voigt profile is recommended when the Doppler line half-width γ D is comparable to the Lorentz half-width γ L [14] , the criterion γ L /γ D < 10 is adopted for selecting the Voigt profile, with the Lorentz profile being used otherwise.A minimum limit of absorption coefficient equals 10 −9 cm −1 is considered as the cut off limit of the line wings.
In order to calculate the absorption coefficient for an arbitrary wavenumber, the contribution of all significant neighboring lines needs to be taken into account.This is done by simply summing up all the contributions of the spectral lines at wavenumber ν, κ ν = i κ ν,i (5) In this work, the line by line absorption spectra of two gases are obtained for the range of 150 cm −1 to 250 0 0 cm −1 .This spectral range is much wider than the one used in development of the atmospheric WSGG model of Bordbar [40] and is wide enough to include the emission power of high temperatures.It for instance accounts for 99.8% of the total balckbody emissive power at 30 0 0 K [33] .
For the sub-atmospheric pressures, a fine resolution of 0 .005 cm −1 is needed to produce accurate absorption spectra of gases.Further refinement beyond this limit has been reported to have negligible effect [33] .It yields to 4,970,0 0 0 evenly spaced spectral intervals of 0.005 cm −1 between 150 cm −1 and 250 0 0 cm −1 .
For atmospheric and higher pressures, the resolution of 0.01 cm −1 is used to obtain the absorption spectra of gases.It is fine enough for these pressures and using higher spectral resolutions has a negligible influence on the results while causing a great increment to the computational cost as shown in the previous reports [57,58] .It yields to 2,485,0 0 0 evenly spaced spectral intervals of 0.01 cm −1 between 150 cm −1 and 250 0 0 cm −1 .
The spectral-line location, intensity, and air and self-broadening are taken from the spectroscopic database HITEMP2010 [48] .For the present work, the profiles of linear spectral absorption coefficient of H 2 O and CO 2 were obtained considering various total pressures, molar fraction ratios and temperatures needed for the WSGG model development.This was done for ten different total pressures (0.1, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0, 30.0, 50.0, and 80.0 atm ) and at fifty-five evenly-spaced temperature values between 300K and 30 0 0 K-i.e., with a step of 50 K .For each total pressure and temperature, thirty-three discrete values of the mole fraction ratio were considered, M r = 0.1, 0.125, 0.25 0.375, 0.5, 0.625, 0.75, 0.875, 1.0, 1.125, 1.25, 1.375, 1.5, 1.625, 1.75, 1.875, 2.0, 2.125, 2.25, 2.375, 2.5, 2.625, 2.75, 2.875, 3.0, 3.125, 3.25, 3.375, 3.5, 3.625, 3.75, 3.875, and 4.0.From these values, the mole fractions of H 2 O and CO 2 (and, consequently, their partial pressures) were calculated by assuming that the gas mixture contains only these two species, following [40] .Hence, the mole fraction of CO 2 is obtained as Although this corresponds to oxygen-fired combustion products, the WSGG model generated from these parameters can be safely applied to air-fired combustion scenarios as well, as demonstrated in [59] for the atmospheric-pressure WSGG model of [40] , which was generated following a similar methodology.The linear absorption spectra of individual gases at each mixture (i.e.represented by a total pressure, a molar fraction ratio and a temperature) were then used to calculate the emissivity of the gas mixtures needed for development of the new pressure dependent WSGG model.
The emissivity of the gas mixture, which is used as the curvefitting parameters for determining the coefficients of the WSGG model (cf.Section 2.2 ) at each pressure, mole fraction ratio and temperature is calculated as where I bν and I b are the spectral and total blackbody radiation intensity, respectively ( and κ ν, CO 2 are the absorption coefficient of water vapor and carbon dioxide, respectively, and L is the path length.For generating the WSGG model, the emissivity is computed for nineteen different path lengths, L = 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 5.0, 10.0, 15.0, 20.0, 30.0, 45.0, and 60.0 m .This, combined to the fifty-five discrete temperature values and thirty-three discrete mole ratio values, yields a LBL-based emissivity database with 34485 points for each one of the ten aforementioned total pressures.

Development of pressure-dependent WSGG model
The WSGG model is a non-gray global spectral model that replaces the absorption spectrum of a given species or mixture by a set of gray gases and a transparent window.Each gray gas i has an absorption coefficient κ i (that does not vary with ν) and covers a fixed although not necessarily continuous, portion ν i of the radiation spectrum.From these considerations, the integration over the spectrum in Eq. ( 6) becomes [60] where N g , a i and κ p,i is the number of gray gases, the weighting factor of each gray gas and its pressure-based absorption coefficient, respectively, and p H 2 O and p CO 2 are the partial pressures of H 2 O and CO 2 .A total of four gray gases ( N g = 4 ) is adopted for the present model, with an additional transparent window denoted as i = 0 .Using larger number of gray gases only marginally improve the accuracy of the model while increasing the CPU cost of RTE solution significantly.
The weighting factors a i in Eq. ( 7) represent the fraction of blackbody energy that lies within ν i .Here, they are described as a fourth order polynomial of the normalized temperature ˆ T = where b i j is the j th degree polynomial coefficient of gas i .Using a normalized temperature instead of the temperature itself in the above equation facilitates the fitting process and is an approach often adopted in the literature (e.g., [3,37,41,51] ).The weighting factor of the transparent window, on the other hand is defined as to satisfy the energy conservation principle, For each molar fraction ratio at each total pressure value, the b i j and κ p,i are found by fitting Eq. ( 7) to the LBL emissivity database across the discrete values of T and L defined in Section 2.1 .The fitting is done with the lsqcurvefit function of MATLAB R2019b, which is a nonlinear least-squares solver as described in [61] .Afterwards, following [40] , the effect of molar fraction ratio in a single set of WSGG model coefficients for each total pressure is included by expressing b i j and κ p,i as in which c i jk and d ik are the constants of the model.For each of the ten total pressures considered here, a different set of constants is obtained.To ease the implementation of the present model, the coefficients of the new WSGG models are given in the supplementary materials of this paper, alongside a simple MATLAB function to implement the model.

The WSGG model and the radiative transfer equation
For participating, non-scattering media, the spectral radiative transfer equation (RTE) is given by [14,16] dI ν ds = −κ ν I ν + κ ν I bν , (12) where I ν is the local spectral radiation intensity, κ ν is the spectral absorption coefficient (for a H 2 O -CO 2 mixture, κ ν = κ ν, H 2 O + κ ν, CO 2 ), and s is the coordinate along the path of radiation propagation.The global solution of the radiative heat transfer entails the integration of the above equation over the spectrum and across all directions.Applying the WSGG model, the total radiation intensity I in a certain direction is determined as a simple summation of the partial intensities I i of each gray gas i (and of the transparent window), i.e., I = ∞ 0 I ν d ν = N g i =0 I i .The partial intensities I i are in turn obtained from the integration of Eq. ( 12) over the region of the spectrum corresponding to the i th gray gas, It has been shown in [60] that the WSGG model can be used alongside any method for the spatial integration of the RTE.In the present study, the discrete ordinates method (DOM) is chosen for this purpose, which solves the RTE for a discrete set of directions and computes the continues integral over all solid angles through a quadrature scheme [62,63] .The cases considered here consists of a one-dimensional medium slab as depicted in Figure 1 , where the participating medium is bounded by two parallel black walls.For this configuration, the DOM determines the partial intensities in the forward and backward directions ( I + i,l and I − i,l , respectively, as also shown in Figure 1 ) from the solution of the following equations where μ l is the cosine of the polar angle θ l in direction l, and x is the spatial position.For black walls with known temperatures, the boundary conditions for these equations are set at the positions x = 0 and x = L (with L being the medium length), respectively: Once Eqs. ( 14) and ( 15) have been solved, the total radiative heat flux and the volumetric radiative heat source at position x ( q r (x ) and S r (x ) , respectively) may be determined as in which N d is half the total number of discrete directions and ω l is the quadrature weight associated to direction l.For the calculations in this study, the values of μ l and ω l are extracted from the quadrature scheme of Lathrop and Carlson [65] , with the weights normalized as to satisfy

Emissivity charts
Although the main advantage of the WSGG model is to accurately account for spectral variation of gas mixtures with a reasonable computational cost, the model has been widely used to calculate the effective absorption coefficient used in gray gas modelingit is, for instance, the base of the main gas radiation property model of Ansys-Fluent [66] .In this kind of gray gas modeling, Beer-Lambert's law is invoked to estimate the effective absorption coefficient from the total emissivity assuming a mean beam length ( L ) as κ = − ln (1 − ε) /L [16] .The emissivity in this formulation is often determined from three-dimensional charts that present ε as a function of the temperature and the path length for a given total pressure and molar fraction ratio.
To assess the accuracy of the WSGG model developed here for such applications, Figure 2 compares the emissivity computed by applying the new model to Eq. ( 7) and the emissivity directly obtained by line-by-line integration of Eq. ( 6) using the absorption spectra extracted from the HITEMP2010 database.For conciseness, only results for typical dry and wet flue gas recycle conditions (i.e., M r = 0 .125 and M r = 1 .0 , respectively) at the smallest and largest pressures considered in this study ( p = 0 . 1 atm and p = 80 atm ) are reported in that figure .For all the cases and across most of the temperature range of 300 K ≤ T ≤ 3000 K, the total emissivities calculated by the present WSGG model are in a good agreement with those determined by the LBL method.However, larger discrepancies are seen at low temperatures in p = 0 . 1 atm , especially for larger path lengths.At this pressure, the emissivity as obtained by the LBL method changes more strongly with temperature for T < 10 0 0 K than anywhere else in the charts particularly for larger lengths.In these low temperatures, the emissivity first decreases with temperature ( 300 K -500 K ), and then increases with temperature until T = 10 0 0 K .The emissivity then decreases with temperature for all the higher temperatures.The larger discrepancies between the emissivity predicted by the WSGG model and LBL method occurring at lower temperatures are explained by the inability of the WSGG model in reproducing the abrupt change of the LBL emissivity.Conversely, for p = 80 atm , the LBL emissivity is less affected by changes of temperature at lower temperatures, and therefore the WSGG model ( Eq. ( 7) ) can more accurately reproduce the emissivities calculated by the LBL method.

Benchmark cases
Three test cases are considered to evaluate the performance of the new WSGG model, consisting of one-dimensional radiative transfer calculations for the domain depicted in Figure 1 , with a medium length L = 1 m .The temperature profile for all cases is the same, given as where ˆ x = x/L, with x denoting the distance from the left wall (cf. Figure 1 ).This is a symmetrical profile, with T = 300 K at the walls and a maximum temperature of 1800 K at the center of the medium (note though that the WSGG model proposed in this paper is applicable for temperatures as high as 30 0 0 K ); similar T -distributions have been previously applied for assessing other WSGG models, e.g., [39,44,50,67] .
The test cases differ between one another by the concentration of the medium, for which three (also symmetrical) profiles are tested, defined by the following mole fraction expressions for H 2 O and CO 2 ( Y H 2 O and Y CO 2 , respectively): Case 1: Case 2: Case 3: Eq. ( 19) illustrates a scenario of air-fuel combustion with M r = 2 , characteristic of the stoichiometric combustion of methane (for this profile, the remainder of the mixture is assumed to be composed of a transparent gas).Eq. ( 20), (21) represent oxy-fuel combustion conditions: the former, with dry flue gas recirculation, where the mole ratio varies between 1/9 and 1/4; and the latter, with wet-flue gas recirculation, in which 0 .5 ≤ M r ≤ 2 .0 .
The solution of the RTE through the discrete ordinates method, as described in Section 3 , is carried out numerically by a finite differencing approach.The medium is discretized in a total 200 grid cells and the number of discrete ordinates for the application of the DOM is set to twelve; further spatial and directional refinements did not significantly impact the results.The reference for assessing the accuracy of the WSGG model is provided by the LBL integration of Eq. ( 12) , which is also solved using the DOM with the same discretization parameters.More details on how the LBL solution is obtained may be found in [39,50,67] .
Figure 3 reports the radiative heat source and radiative heat flux for Test Cases 1 to 3 for a medium at atmospheric pressure, as computed by LBL integration and by the WSGG model proposed in this paper.For comparison purposes, the results of the WSGG formulation of Shan et al. [51] -which is the only other M r -varying WSGG correlation available in the literature that is applicable to high-pressures, cf.Table 1 -are also included in the figure.Differently from the present model, the one by Ref. [51] incorporates a dependence of pressure into the gray gas weighting and absorption coefficients, but it was developed for several fixed M r values; to account for cases with varying M r , a stepwise approach was proposed, in a similar way to some atmospheric-pressure WSGG formulations [36,52] .The problem with this approach is that it may introduce discontinuities in the results, leading to non-physical distributions of S r , as it will be shown later.Thus, as an attempt to circumvent this issue, an additional application of the model of Shan et al. [51] has been carried out here, where the absorption coefficient and polynomial terms that compose the weighting coefficient are linearly interpolated between their values proposed at the fixed M r points; a similar approach has been adopted by Kez et al. [68] for testing the stepwise WSGG model of [52] .
Both the new WSGG model and the one by Ref. [51] are able to capture the overall spatial behavior of S r and q r in Figure 3 .However, while Shan's model even outperforms the present model for the prediction of the radiative heat flux in Case 1, where the mole ratio does not vary (notice that for this case the stepwise and interpolated approaches for that model yield identical results, so only the former is reported), the same is not verified for neither S r nor q r for Cases 2 and 3.In those cases, some abrupt changes in the spatial distribution of the radiative heat source can be observed for the application of the model of Shan et al. that uses the standard stepwise approach for determining κ p,i and a i ; these changes in the S r slope occur precisely at points where the mole ratio transitions from one of the WSGG coefficients set in Ref. [51] to another.Similar findings [68] have been reported for other WSGG formulations that use a stepwise procedure to deal with M rvarying scenarios.Linearly interpolating Shan's coefficients eliminates this issue, but still the accuracy of that model is less than that of the model presented in this paper.This is further demonstrated in Table 2 , that reports the maximum ("max") and medium-averaged ("avg") normalized errors for each model.The normalized error is defined as φ where φ is either S r or q r ; φ LBL and φ WSGG are the local values of φ obtained by the solutions with the LBL method and the WSGG model, respectively; and max (| φ LBL | ) is the maximum absolute value of φ LBL in the medium.While Shan's WSGG model shows comparable (for S r ) or smaller (for q r ) errors than the present model for Case 1, it is clear that the new model performs better for Cases 2 and 3, which correspond to varying-M r scenarios.The improvement over the standard stepwise formulation of Shan et al. [51] is especially significant for the maximum errors, which go from ranging between 15% and 22% to, at worst, 17%.Moreover, although the maximum errors of Shan's model reduce (at least for S r ) if the interpolation approach is adopted instead, its average errors are still larger than those of the new model.Table 2 includes the errors of the WSGG model of Bordbar et al. [40] .Compared to that model [40] , the new model fares either as well (for Case 1) or better (for Cases 2 and 3).Particularly for Case 2, the errors of the new model are almost half of that of the older model.Nevertheless, it is interesting to note that, although it was developed for atmospheric conditions, the model of Bordbar et al. [40] has a fairly good accuracy for Case 1, even outperforming Shan's model (this is also true for Case 3).
Results for Cases 1 to 3 with a total pressure of 30 atm are depicted in Figure 4 .For this higher pressure, the improvement in accuracy offered by the present model becomes more evident.Its errors do not surpass 10% for any of the test cases, and, on aver-age, they are always less than 5%-for contrast, the average errors of Shan's model are, at best, above 6% (see Table 2 ).Furthermore, for Cases 2 and 3 (especially for the former), the non-physical spatial distribution of S r attained with the stepwise approach of Shan's model are more clearly noticeable for p = 30 atm than for p = 1 atm , and result in local errors as high as 60%.Once again, while adopting a linear interpolation to deal with the varying mole ratio eliminates this issue, it still leads to errors considerably larger than those of the present model.
One possible reason for the overall lower performance of Shan's model could be the total pressure fitting that is performed on their correlations, which is not performed on the present model.The  deviations caused by this additional fitting, even if small, might be contributing to the differences between the two models.Furthermore, Shan's coefficients are based on the EM2C narrow-band emissivities, which have their own deviations when compared to LBL emissivities.Even though the SNB usually presents great agreement with the LBL solution, the fact that the database from EM2C is based on absorption spectra that are different than the ones considered in this study might also be a relevant factor to explain the relatively lower accuracy of Shan's model.
A medium at sub-atmospheric pressure is considered next.Figure 5 plots the resulting S r and q r distributions for Cases 1 to 3 for a medium at a total pressure p = 0 .5 atm ; because no previous WSGG formulation is available for sub-atmospheric pressures, only results of the present WSGG model and of the LBL integration method are reported.For this pressure, the model performs the best for Case 1, where its average errors are below 4% for both S r and q r , as conveyed by Table 2 .For Cases 2 and 3, the errors are larger, especially for the radiative heat flux close to the walls, reaching almost 20% for Case 2. Nonetheless, on average, for the p = 0 .5 atm cases the errors of the new model range between 5% and 8%.

Performance of the model across its pressure range
To further assess the performance of the new model, the temperature and species concentration profiles corresponding to Cases 1 to 3 have been simulated for different total pressures of the medium.The values chosen for p correspond to the discrete values for which the set of new WSGG correlations were developed, as discussed in Section 2 , and range from 0.1 atm to 80 atm .
Figure 6 reports the medium-averaged errors of the model for S r and q r as a function of the total pressure.Starting at p = 10 atm , the accuracy of the model does not vary much with increasing pressures, with the average errors never surpassing 5% (the exception being in the radiative heat flux for Case 1, where the average error about 6% for p = 80 atm ).For sub-atmospheric pressures, the model also fares well, although for Case 2 ( q r ) avg reaches 9.5% for p < 1 .0 atm .Note that the main difference between these three cases are in how the molar fraction ratio of H 2 O to CO 2 (i.e.M r ) varies along the length of the 1D benchmarks.In case 1, the M r does not vary and always equals two.This makes the accuracy of the WSGG model better in case 1 than the other two cases in which M r varies and therefore the inaccuracy of the WSGG model is more due to the second stage fitting over M r , i.e.Eq. (11) .The accuracy improvement seen in case 2 and 3 with increasing the pressure is guesstimated to be because of the smoother absorption spectra of the gas mixtures at high pressures.Due to simpler form of absorption spectra, the WSGG seems to better simulate the spectral radiation of the gas mixtures at higher pressures.

Conclusions
While recent line-by-line based WSGG models such as [40] and [38] show promising computational performance in modeling spectral radiation in large atmospheric combustors [22,68] , their accuracy is considerably lower when they are used for modeling non-atmospheric combustion systems [21] .With this in mind, a new WSGG model was developed in the present paper for the treatment of H 2 O -CO 2 mixtures with varying mole fraction ratio, applicable to temperatures between 30 0K and 30 0 0 K and to total pressures of 0.1 atm to 80 atm .The model was constructed by fitting total emissivity data to the high-resolution spectral database HITEMP2010; a very good agreement was found when comparing the emissivity charts resulting from the model to those computed by line-by-line integration of the absorption spectrum.The performance of the new model was further analyzed by applying it to several one-dimensional benchmarks representing various levels of complexity in temperature and gas composition profiles.Once again, the model consistently showed a satisfactory agreement with the predictions of LBL calculations, even outperforming the only existing WSGG model applicable to the conditions studied in this work [51] .In summary, the average error in the radiative source calculated by the new model at various pressures was always less than 5%, and the worst error in the radiative heat flux was about 10%.

Declaration of Competing Interest
Reffering to manuscript entitled as "Pressure-dependent weighted-sum-of-gray-gases models for heterogeneous CO2-H2O mixtures at sub-and super-atmospheric pressure", we wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed.We further confirm that the order of authors listed in the manuscript has been approved by all of us.We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property.In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property.We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office).He is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs.We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author.

Fig. 1 .
Fig. 1.Schematic representation of the one-dimensional domain for the RTE solution with the WSGG model and the DOM (adapted from [64] ).

Fig. 2 .
Fig. 2. Emissivity charts for typical dry and wet flue gas recycle conditions obtained by LBL integration and by the new WSGG model.

Fig. 3 .
Fig. 3. Results of the test cases with p = 1 atm .

Fig. 4 .
Fig. 4. Results of the test cases with p = 30 atm .

Fig. 6 .
Fig. 6.Average errors in the radiative heat flux and radiative heat source for the new WSGG model as a function of the total pressure of the medium.

Table 1
Summary of the different existing formulations of the WSGG model developed for high-pressure applications.

Table 2
[40,51] and average (in parenthesis) normalized errors of the new model and the models of Refs.[40,51].
a Interpolated