Augustin Guibaud

Jean-Louis Consalvi
email: jean-louis.consalvi@univ-amu.fr

Jean-Marie Citerne

Guillaume Legros

Pressure Effects on the Soot Production and Radiative Heat Transfer of Non-Buoyant Laminar Diffusion Flames Spreading in Opposed Flow over Insulated Wires

This paper investigates experimentally and numerically pressure effects on soot production and radiative heat transfer in non-buoyant opposed-flow flames spreading over wires coated by Low Density PolyEthylene (LDPE). Experiments, conducted in parabolic flights, consider pressure levels ranging from 50.7 kPa to 121.6 kPa and an oxidizer flowing parallel to the wire's axis at a velocity of 150 mm/s and composed of 20% O2/80% N2 in volume. The numerical model includes a detailed chemistry, a two-equation smoke-point based soot production model, a radiation model coupling the Full-Spectrum correlated-k method with the finite volume method and a simple degradation model for LDPE. An analysis of the experimental data shows that the spread rate, the pyrolysis mass flow rate, and the residence time for soot formation are independent of pressure whereas the soot formation rate is third-order in pressure. The model reproduces quantitatively the effects of pressure on soot production and captures the transition from non-smoking to smoking flames. The radiant fraction increases with pressure because of

an enhancement in soot radiation whereas the contribution of radiating gases remains approximately constant over the range of pressures considered. In addition, gas radiation dominates at pressure lower than 75 kPa whereas soot radiation prevails at higher-pressure levels. Consistently with the data obtained at normal gravity, the smoke-point transition is found to occur for a radiant fraction of about 0.3 and the soot oxidation freezing temperature is estimated in the range 1350-1450K. Eventually, whatever the pressure considered, the surface re-radiation from the wire is higher than the incident radiative flux from the flame to the surface along the entire wire. This shows that radiative heat transfer contributes negatively to the heating of the unburnt LDPE and to the heat balance along the pyrolysing surface. Δℎ 𝑐 heat of combustion [J kg -1 ]

 absorption coefficient [m -1 ] 𝜅 𝑃 Planck-mean absorption coefficient [m -1 ] 𝜆 thermal conductivity [W•m -1 •K -1 ] 𝜇 dynamic viscosity [kg•m -1 •s -1 ]  density [kg•m -3 ] 𝜎 Stefan-Boltzmann constant [W m -2 K -4 ]
𝜒 𝑅 radiant fraction [-] 𝜒 𝑅,𝑔 gas contribution to the radiant fraction [-] 𝜒 𝑅,𝑠 soot contribution to the radiant fraction [-] 𝜔̇𝑐 𝑜𝑎𝑔 reaction rate for soot coagulation [part•m -3 •s -

Introduction

Unexpected overheating of wires by electrical current overshoots has been identified as a primary cause of fire initiation and growth in a space vehicle. This has motivated a significant amount of experimental [START_REF] Greenberg | The USML-1 Wire insulation Flammability Glovebox Experiment[END_REF]- [START_REF] Guibaud | Experimental evaluation of flame radiative feedback: methodology and application to opposed flame spread over coated wires in microgravity[END_REF] and numerical [START_REF] Takahashi | Study on unsteady molten insulation volume change during flame spreading over wire insulation in microgravity[END_REF], [START_REF] Guibaud | Soot production and radiative heat transfer in opposed flame spread over a polyethylene insulated wire in microgravity[END_REF] studies related to laminar flame spread over thin electrical wires in microgravity. This motivation has been reinforced by the inherent differences between non-buoyant and buoyant laminar diffusion flame structures [START_REF] Urban | Flame spread: Effects of microgravity and scale[END_REF] that do not allow the conclusions drawn at earth gravity to be extended to microgravity. Ambient conditions, such as pressure, oxygen content, and slow forced flow due to ventilation are expected to affect flame spread characteristics. Among these parameters, oxygen content and ambient pressure are intimately related in space exploration applications.

Ranging from conservative Earth-like conditions to ambitious low-pressure and high oxygen content environment, past spacecraft designs have incorporated one of the following three sets of pressure and oxygen content: i) pure molecular oxygen (𝑥 𝑂 2 = 1) at a reduced pressure, 𝑃, [START_REF] Diamant | Past and present environmental control and life support systems on manned spacecraft[END_REF]. In the near future, exploration atmosphere is likely to be an oxygen-enriched mixture at low pressure, with current target for the Orion Multi-Purpose Crew Vehicle of 𝑥 𝑂 2 = 0.34 and 𝑥 𝑁 2 = 0.66 at 𝑃 = 56.5 𝑘𝑃𝑎 [START_REF] Urban | Flame spread: Effects of microgravity and scale[END_REF]. It should be pointed out that none of these atmospheric conditions are fire safe since the flammability limit of most of hydrocarbons is about 15% in terms of O2 mole fraction regardless of pressure. Fire incidents have been documented aboard Apollo 13 [9], Salyut-1 [START_REF] Ivanovich | Salyut -The First Space Station: Triumph and Tragedy[END_REF], [START_REF] Siddiqi | Challenge to Apollo: the Soviet Union and the space race, 1945-1974[END_REF], Salyut-6 [START_REF] Oberg | Star-crossed orbits: inside the US-Russian space alliance[END_REF], the Orbiter fleet [START_REF] Friedman | Risks and issues in fire safety on the space station[END_REF], and Mir [START_REF] Baker | The Story of Manned Space Stations: An Introduction[END_REF]. In addition, the vast majority of incidents report surprising large amounts smoke [START_REF] Garber | Fire and Controversy January 12[END_REF], which raises doubts regarding the future safety of astronauts onboard the Orion spacecraft in the context of long-range missions. Consequently, understanding how ambient parameters affect the onset of smoke release is crucial towards the development of a safe space exploration framework.

of
In order to precisely break down the mechanisms responsible for the transition from nonsmoking to smoking-flames, a special emphasis must be put on soot production and the related radiative heat transfer processes. Previous studies considering non-buoyant axisymmetric laminar diffusion flames fueled by gaseous hydrocarbons showed that, as observed at normal gravity [START_REF] Karatas | Soot formation in high pressure laminar diffusion flames[END_REF]- [START_REF] Daca | Soot formation characteristics of diffusion flames of methane doped with toluene and n-heptane at elevated pressures[END_REF], the soot production and therefore soot radiation increase with pressure [START_REF] Urban | Structure and soot properties of nonbuoyant ethylene/air laminar jet diffusion flames[END_REF], [START_REF] Urban | Smoke-point properties of nonbuoyant round laminar jet diffusion flames[END_REF]. A consequence of this increase in fuel sooting propensity with pressure is that the laminar smoke point (LSP) flame length is inversely proportional to pressure [START_REF] Urban | Smoke-point properties of nonbuoyant round laminar jet diffusion flames[END_REF]. The mechanisms leading to the SP were found similar at both normal and microgravity and were identified as the quenching of the soot oxidation process at the flame tip owing to the radiative cooling mainly caused by soot radiation. However, the SP characteristics in microgravity were found different from those observed at earth gravity. First, non-buoyant SP flame lengths are significantly shorter for comparable conditions [START_REF] Urban | Smoke-point properties of nonbuoyant round laminar jet diffusion flames[END_REF]. Moreover, the SP radiant fraction and the soot oxidation freezing temperature for microgravity were reported to be in the range 0.4-0.6 [START_REF] Urban | Structure and soot properties of nonbuoyant ethylene/air laminar jet diffusion flames[END_REF] and 1000 K [START_REF] Urban | Smoke-point properties of nonbuoyant round laminar jet diffusion flames[END_REF], respectively, as compared to 0.3 [START_REF] Markstein | Relationship between smoke point and radiant emission from buoyant turbulent and laminar diffusion flames[END_REF] and the range 1300-1450 K [START_REF] Markstein | Radiant emission and absorption by ethylene and propylene diffusion flames[END_REF]- [START_REF] Bonnety | Probing the local radiative quenching during the transition from a non-smoking to a smoking laminar coflow ethylene/air non-premixed flame[END_REF] at normal gravity.

In addition to these gas-phase phenomenological discrepancies, alterations in flame spread rate are well documented in various configurations in the absence of buoyancy [START_REF] Kikuchi | Experimental study on flame spread over wire insulation in microgravity[END_REF], [START_REF] Urban | Flame spread: Effects of microgravity and scale[END_REF], [START_REF] Olson | Flame spread over thin fuels in actual and simulated microgravity conditions[END_REF]. Since flame spread controls the fuel pyrolysis and, in turn, the flame geometry and residence time, these modifications increased doubts in the ability of standard tests conducted at normal gravity to successfully forecast smoking properties of a given material. Investigations of SP mechanisms over a spreading flame thus combine both effects to provide a holistic perception of the definite effects of flow conditions.

The present paper focuses on the soot production, the SP mechanisms and radiative structures of laminar diffusion flames spreading in an opposed flow configuration over idealized electrical wires. The main advantage of the wire configuration is that its 2D axisymmetric geometry allows the BMAE (Broadband Modulated Absorption Emission) technique, cautiously developed by the authors [START_REF] Guibaud | Broadband modulated absorption/emission technique to probe sooting flames: Implementation, validation, and limitations[END_REF], [START_REF] Guibaud | Experimental evaluation of flame radiative feedback: methodology and application to opposed flame spread over coated wires in microgravity[END_REF], to be implemented, which leads to the concomitant measurements of soot volume fraction and temperature in the spreading flame. The wires consist of a Nickel-Chrome (NiCr) metallic core coated with Low Density PolyEthylene (LDPE) and differ from the non-flammable polyimide wire insulation (MIL-W-81381) mostly employed in spacecraft [START_REF] Stavnes | Operational Environments for Electrical Power Wiring on NASA Space Systems[END_REF]. However, although LDPE coated wires are not employed in space vehicles as such, their consideration is relevant for several reasons. First, it corresponds to an international target configuration to investigate fundamentally the flammability properties of electrical wires. Their use has emerged with the work of Bakham et al. [START_REF] Bakhman | Burning of polymeric coatings on copper wires and glass threads: I. Flame propagation velocity[END_REF] and, over the last twenty years, a significant amount of experimental studies, mainly related to fire safety in spacecraft, have adopted the PE coated wire configuration [START_REF] Fujita | Effect of low external flow on flame spread over polyethylene-insulated wire in microgravity[END_REF], [START_REF] Guibaud | Experimental evaluation of flame radiative feedback: methodology and application to opposed flame spread over coated wires in microgravity[END_REF], [START_REF] Friedman | Fire safety in spacecraft[END_REF]- [START_REF] Nagachi | Can a spreading flame over electric wire insulation in concurrent flow achieve steady propagation in microgravity?[END_REF]. Second, it produces soot volume fraction levels above the detection threshold of the BMAE over the range of pressure considered in the present study. The lower limit of the pressure range was selected to be above this threshold and the range was calibrated to observe the transition from a nonsmoking flame to a smoking one. Finally, PE is a potential spacecraft material considered for space radiation shielding [START_REF] Harrison | Polyethylene/boron nitride composites for space radiation shielding[END_REF], and hence requires a careful evaluation of its flammability properties. The present work is organized as follows. The second section presents the experimental setup and optical diagnostics. The numerical model is presented in section 3. The experimental and numerical results are discussed in section 4 whereas section 5 summarizes the main conclusions of the paper.

Experiments

Experiments were conducted in parabolic flights. The experimental setup and the optical diagnostics are detailed in Ref. [START_REF] Guibaud | Experimental evaluation of flame radiative feedback: methodology and application to opposed flame spread over coated wires in microgravity[END_REF] and are only briefly described here. The experimental setup consists of a cylindrical combustion chamber with an inner diameter of 190 mm. Cylindrical wires of length 150 mm, consisting of a 0.5 mm diameter NiCr core coated by a 0.3 mm thick LDPE insulation, are placed along the central axis of the chamber. Laminar oxidizer flows of velocity 150mm/s, parallel to the sample axis and composed of 20% O2/80% N2 in volume, were generated. The pressure was varied between 50.7 and 121.6 kPa. The samples were ignited using a hot Kanthal wire, resulting in an opposed-flow flame spread.

Images were captured using a JAI AT-140CL digital tri-CCD camera, equipped with a telecentric lens to restrict the light collection to beams parallel to the optical axis. With this arrangement, the spatial resolution of the projected data is 76 m over the two spectral bands of interest for the BMAE, ranging from 480 nm to 600 nm (green) and from 570 nm to 700 nm (red), respectively. Images were acquired at a rate of 39.06 fps and a LEDs backlight behind the samples is alternatively set on and off.

Flame spread rate, pyrolysis mass flow rate, SVF and temperature are determined using image processing once a steady flame spread is reached. The flame is assumed to spread at a steady rate once the dimensions of the characteristic molten insulation droplet (see Fig. 1), the visible flame length, and the rate of the flame front displacement reach a steady state value [START_REF] Guibaud | Experimental evaluation of flame radiative feedback: methodology and application to opposed flame spread over coated wires in microgravity[END_REF]. The flame spread rate, 𝑢 𝑝 , is evaluated by tracking the time evolution of the luminous flame front.

The pyrolysis rate, 𝑚̇𝑝, is then evaluated from 𝑢 𝑝 assuming a constant LDPE density of 𝜌 𝑃𝐸 = 920 kg/m 3 .

SVF and temperature fields are obtained using the B-MAE technique as extensively outlined in

Ref. [START_REF] Guibaud | Broadband modulated absorption/emission technique to probe sooting flames: Implementation, validation, and limitations[END_REF]. As discussed in the introduction, the implementation of the BMAE takes advantage of the axisymmetric geometry of the present configuration. These fields can be extracted from a set of four images: the flame with backlight, the backlight alone, the flame alone, and the background noise. The technique assumes that soot radiates within the Rayleigh regime. The soot absorption coefficient is computed using the refractive indexes provided by Chang and

Charalampopoulos [START_REF] Chang | Determination of the wavelength dependence of refractive indices of flame soot[END_REF]. The transmission fields over each spectral range, obtained by subtracting images with and without backlight, allows to recover the SVF. As the information is integrated over line-of-sights, a deconvolution procedure, involving an onion-peeling method combined with a Tikhonov regularization, is required. The frames captured over the two spectral ranges in the absence of backlight are then processed to retrieve the temperature. Soot reabsorption is taken into account in the onion-peeling deconvolution process since it is nonnegligible over the spectral ranges involved in the BMAE. This technique allows SVF and temperature to be recovered with accuracies under 1 ppm and 100 K, respectively [START_REF] Guibaud | Experimental evaluation of flame radiative feedback: methodology and application to opposed flame spread over coated wires in microgravity[END_REF], [START_REF] Guibaud | Broadband modulated absorption/emission technique to probe sooting flames: Implementation, validation, and limitations[END_REF]. It should be noticed that the chemiluminescence by species other than soot is neglected within the spectral ranges probed (500 -700 nm). This assumption is backed by the high-energy broadband nature of soot emission, while interfering gaseous species in this region of the visible spectrum such as C2 marginally contribute to the overall radiation. This is supported by the fine agreement between the distributions of soot temperature as obtained by the BMAE and the Spectral Modulated Absorption Emission (SMAE) techniques [START_REF] Guibaud | Broadband modulated absorption/emission technique to probe sooting flames: Implementation, validation, and limitations[END_REF]. Indeed, the narrow bands used by the SMAE, i.e. 645 ± 5 nm and 785 ± 5 nm, were selected to get signals freed from the influence of the aforementioned interfering species.

Numerical model

Governing equations

The model solves the steady-state governing equations of both gaseous and solid phases in a flame-fixed axisymmetric coordinate system [START_REF] Guibaud | Soot production and radiative heat transfer in opposed flame spread over a polyethylene insulated wire in microgravity[END_REF].

A detailed description of the solid phase degradation model can be found in Ref. [START_REF] Guibaud | Soot production and radiative heat transfer in opposed flame spread over a polyethylene insulated wire in microgravity[END_REF] and only a summary is given here. The study focuses on gas phase processes with a special emphasis on soot production and radiative heat transfer. Consequently, the solution of the conjugated heat transfer problem at the gas/solid interface is only needed to specify proper boundary conditions for the gas phase. The problem is simplified by specifying the spread rate, the pyrolysis rate and shape of the molten LDPE droplet as inputs from the steady spreading experimental data.

Based on experimental observations, the molten balls were assumed to have the same shape and size within the range of pressures investigated in the present study (see Fig. S2 in the supplementary materials). The condensed phase is then divided into three regions (see Figs.

1
and S2 for illustration). The first region is the unburnt insulated wire located ahead of the pyrolysis front (𝑧 ≤ 0). The second region is the molten ball where pyrolysis is assumed to occur. The fuel injection velocity profile, 𝑢 𝐹 (𝑧), is inferred from the measured fuel mass flow rate, 𝑚̇𝑝, the ball surface, 𝑆 𝑏 , and the stand-off distance, 𝛿(𝑧). The analysis, described in Ref.

[6], is based on the classical result of the reactive-laminar boundary layer theory, showing that the pyrolysis mass flow rate per unit area, 𝑚̇𝑝 ′′ , is inversely proportional to the stand-off distance [START_REF] Emmons | The Film Combustion of Liquid Fuel[END_REF]. The third region is the bare nickel-chrome core (𝑧 > 𝑧 𝑏 ) where 𝑧 𝑏 denotes the delimitation between the molten ball and the bare core. The axisymmetric heat transfer equation for both the metal core and LDPE is solved without introducing the thermally thin approximation, based on past observation of radial thermal gradient in this configuration [START_REF] Guibaud | Soot production and radiative heat transfer in opposed flame spread over a polyethylene insulated wire in microgravity[END_REF]. Both melting and pyrolysis processes are assumed to be phase transition. The PE can be into three states depending on the local temperature. First, in a solid phase if the temperature is lower than the melting temperature, 𝑇 𝑚𝑒𝑙𝑡 . Second, at the transition if the temperature is equal to 𝑇 𝑚𝑒𝑙𝑡 . Third, in the molten phase if the temperature is higher than 𝑇 𝑚𝑒𝑙𝑡 . The melting point of LDPE is assumed to occur at 𝑇 𝑚𝑒𝑙𝑡 =403 K [START_REF] Guibaud | Experimental evaluation of flame radiative feedback: methodology and application to opposed flame spread over coated wires in microgravity[END_REF], [START_REF] Guibaud | Soot production and radiative heat transfer in opposed flame spread over a polyethylene insulated wire in microgravity[END_REF] and different thermal properties are used for the solid and the molten LDPE [START_REF] Guibaud | Soot production and radiative heat transfer in opposed flame spread over a polyethylene insulated wire in microgravity[END_REF]. During the melting stage, the thermal properties are computed using a mass-weighted average between the properties of solid and those of the molten LDPE [START_REF] Guibaud | Soot production and radiative heat transfer in opposed flame spread over a polyethylene insulated wire in microgravity[END_REF]. The pyrolysis is assumed to occur at the ball surface at a fixed temperature, 𝑇 𝑝 =760 K. All the thermal constants for the LDPE, including density, conductivity and heat capacity of both virgin and molten LDPE, temperature and heat of melting and pyrolysis temperature were taken from thermal analysis [START_REF] Guibaud | Experimental evaluation of flame radiative feedback: methodology and application to opposed flame spread over coated wires in microgravity[END_REF], [START_REF] Guibaud | Soot production and radiative heat transfer in opposed flame spread over a polyethylene insulated wire in microgravity[END_REF], [38]. The emissivity was set equal to 1 for both solid and molten LDPE as well as for the NiCr along the bare wire. This simplification ignores the spectral dependency of the emissivity as well as its alteration resulting from the possible deposit of soot.

The numerical model solves the Navier-Stokes equations and transport equations for gas-phase species mass fractions and energy. More details concerning the numerical methods and the strategy for handling the stiffness of the equation system are given in Ref. [START_REF] Guo | Numerical study on the influence of hydrogen addition on soot formation in a laminar ethylene-air diffusion flame[END_REF].

A detailed modeling of diffusion flames fueled by gaseous pyrolysis products released from solids is challenging. In the case of LDPE, the composition of the gaseous pyrolysis products is complex and temperature-dependent [START_REF] Lovett | Ultrapyrolytic upgrading of plastic wastes and plastics/heavy oil mixtures to valuable light gas products[END_REF]. In addition, the compositions reported in the literature may not be accurate enough, especially in terms of soot precursors, to be used as inputs data for detailed chemical mechanisms. Finally, detailed chemical mechanisms for such mixtures may be not available. The strategy adopted in the present study is inspired from that used by Zhang et al. [START_REF] Zhang | A burner to emulate condensed phase fuels[END_REF] and Markan et al. [START_REF] Markan | A Burning Rate Emulator (BRE) for study of condensed fuel burning in microgravity[END_REF] to develop the Burning Rate Emulator (BRE)

to simulate the burning of condensed fuels. These authors identified four key parameters that have to be preserved to mimic the burning of solids: the heat of combustion (or similarly the oxygen-fuel mass ratio) to preserve the flame structure, the laminar smoke point height to preserve the fuel sooting propensity, the surface temperature to preserve surface heat loss by re-radiation and the heat of gasification to preserve the pyrolysis mass flow rate. As discussed previously, the pyrolysis mass flow rate and the surface temperature of the pyrolysing surface were imposed in the present study from experimental data. Pure ethylene is a relevant candidate to represent the pyrolysis products of LDPE since both species exhibit very similar heat of combustion with a discrepancy lower than 10% and the same oxygen-fuel mass ratio [START_REF] Khan | Combustion Characteristics of Materials and Generation of Fire Products[END_REF]. This assumption, adopted in previous studies, is supported by the good agreement observed in Ref.

[6] between predicted and measured temperature and stand-off distance. Finally, the fuel sooting propensity of LDPE will be preserved by using its smoke point length in the sooting model (see Section 3.2). The oxidation of ethylene is modeled using the full chemical kinetic scheme developed by Qin et al. [START_REF] Qin | Combustion chemistry of propane: a case study of detailed reaction mechanism optimization[END_REF].

Soot production model

A modified two-equation soot production model, based on transport equations for soot number density per unit mass of mixture (𝑁 𝑠 ) and soot mass fraction (𝑌 𝑠 ), is proposed. The source term, 𝜔̇𝑌 𝑠 = 𝜔̇𝑠 𝑓 -(𝜔̇𝑂 2 + 𝜔̇𝑂 𝐻 )𝐴 𝑆 , accounts for soot formation and oxidation by O2 and OH, with 𝐴 𝑆 being the soot surface area.

The LSP based soot formation (SF) model proposed by Lautenberger et al. [START_REF] Lautenberger | A simplified model for soot formation and oxidation in CFD simulation of non-premixed hydrocarbon flames[END_REF] is used to compute 𝜔̇𝑠 𝑓 = 𝑓 𝑠𝑓 (𝜉)𝑔 𝑠𝑓 (𝑇), where 𝜉 and 𝑇 represent mixture fraction and temperature, respectively. 𝑓 𝑠𝑓 and 𝑔 𝑠𝑓 are cubic functions whose coefficients are determined by specifying the value and slope of the polynomials at the peak and at the lower and higher limits of the SF process. The model constants, calibrated in Ref. [START_REF] Lautenberger | A simplified model for soot formation and oxidation in CFD simulation of non-premixed hydrocarbon flames[END_REF], are kept unchanged in the present study.

The model is generalized to an arbitrary fuel by relating the peak of SF rates, 𝜔̇𝑠 𝑓,𝑃 , to the LSP height [START_REF] Lautenberger | A simplified model for soot formation and oxidation in CFD simulation of non-premixed hydrocarbon flames[END_REF]:

𝜔̇𝑠 𝑓,𝑃 = 1.1 𝑙 𝑠𝑝,𝐶 2 𝐻 4 𝑙 𝑠𝑝,𝑃𝐸 ( 𝑊 𝐶 2 𝐻 4 𝑊 𝑃𝐸 ) ( 𝑃 𝑃 0 ) 𝑛 (1)
where 𝑙 𝑠𝑝,𝐶 2 𝐻 4 and 𝑙 𝑠𝑝,𝑃𝐸 are the LSP heights of ethylene and PE equal to 0.106m and 0.045m, respectively [START_REF] Khan | Combustion Characteristics of Materials and Generation of Fire Products[END_REF]. 𝑃 is the local pressure whereas 𝑃 0 is the reference one (101.3kPa). The exponent 𝑛 was set equal to 2 in the initial model formulation [START_REF] Lautenberger | A simplified model for soot formation and oxidation in CFD simulation of non-premixed hydrocarbon flames[END_REF] based on the analysis provided in Ref. [START_REF] De Ris | Radiation Fire Modelling[END_REF] from the early experimental data of Refs. [START_REF] Schalla | Formation and combustion of smoke in laminar flames[END_REF] and [START_REF] Flower | Soot production in axisymmetric laminar diffusion flames at pressures from one to ten atmospheres[END_REF]. However, recent experiments in laminar axisymmetric coflow diffusion flames at normal gravity showed that this exponent depends on both the fuel and pressure range [START_REF] Karatas | Soot formation in high pressure laminar diffusion flames[END_REF]- [START_REF] Daca | Soot formation characteristics of diffusion flames of methane doped with toluene and n-heptane at elevated pressures[END_REF]. In the present study, it will be calibrated from an analysis of the experimental data (see Section 4.1). 𝑊 𝐶 2 𝐻 4 and 𝑊 𝑃𝐸 represent the molecular weights of ethylene and pyrolysis products of PE, respectively. The accepted vaporization mechanism of PE considers that it vaporizes as higher molecular weight oligomers than as monomers [START_REF] Khan | Combustion Characteristics of Materials and Generation of Fire Products[END_REF]. As discussed in Section 3.1, the composition of the pyrolysis products and, in turn, 𝑊 𝑃𝐸 are tricky to estimate. As a consequence, the ratio 𝑊 𝐶 2 𝐻 4 𝑊 𝑃𝐸 ⁄ was disregarded in Eq. ( 1) and the constant 1.1 was recalibrated to match the experimental SVF peak for the flame at P=101.3kPa. The calibrated constant was found equal to 0.5.

The soot oxidation rates by OH and O2 optimized by Guo et al. [START_REF] Guo | Optimized rate expressions for soot oxidation by OH and O2[END_REF] were considered. This amounts to model 𝜔̇𝑂 𝐻 by the Fenimore and Jones expression [START_REF] Fenimore | Oxidation of soot by hydroxyl radicals[END_REF] with a revised collision efficiency of 0.10 whereas 𝜔̇𝑂 2 is computed as 𝜔̇𝑂 2 = 𝐴 𝑂 2 𝑃 𝑂 2 𝑇 -0.5 𝑒𝑥𝑝(-𝐸 𝐴,𝑂 2 𝑅 𝑢 ⁄ 𝑇) with 𝑃 𝑂 2 and 𝑅 𝑢 being the partial pressure of O2 and the universal gas constant, respectively.

The source term, 𝜔̇𝑁 𝑆 , for the soot number density is given by 𝜔̇𝑁 𝑆 = 𝑁 𝐴 𝑁𝐶 𝑚𝑖𝑛 ⁄ 𝜔̇𝑛 -𝜔̇𝑐 𝑜𝑎𝑔

where 𝑁 𝐴 and 𝑁𝐶 𝑚𝑖𝑛 =60 are the Avogadro number and the number of carbon atoms in the incipient soot particle [START_REF] Lindstedt | Simplified soot nucleation and surface growth steps for non-premixed flames[END_REF], respectively. The acetylene-based inception model developed by

Lindstedt [START_REF] Lindstedt | Simplified soot nucleation and surface growth steps for non-premixed flames[END_REF] is revised by multiplying the initial formulation by 𝑙 𝑠𝑝,𝐶 2 𝐻 4 𝑙 𝑠𝑝,𝑃𝐸 ⁄ to account for the difference in sooting propensity between ethylene and PE:

𝜔̇𝑛 = 2𝑘 1 (𝑇) 𝑙 𝑠𝑝,𝐶 2 𝐻 4 𝑙 𝑠𝑝,𝑃𝐸 ⁄ [𝐶 2 𝐻 2 ] (2)
[𝐶 2 𝐻 2 ] is the acetylene concentration. The kinetic parameters, 𝑘 1 (𝑇), are those provided by Lindstedt [START_REF] Lindstedt | Simplified soot nucleation and surface growth steps for non-premixed flames[END_REF]. The coagulation rate, 𝜔̇𝑐 𝑜𝑎𝑔 , is also computed as proposed by Lindstedt [START_REF] Lindstedt | Simplified soot nucleation and surface growth steps for non-premixed flames[END_REF].

Radiative model

The Full-Spectrum Correlated-K (FSCK) is used as a gas-soot radiative property model. The spectral lines for CO2 and H2O are taken from HITEMP 2010 [START_REF] Rothman | HITEMP: The High-Temperature Molecular Spectroscopic Database[END_REF] whereas the refractive indices reported in Ref. [START_REF] Chang | Determination of the wavelength dependence of refractive indices of flame soot[END_REF] are used to compute the soot absorption coefficient.

The FS RTE is expressed as [START_REF] Modest | Assembly Full Spectrum k-Distribution from a Narrow Band Database: Effects of Mixing Gases, Gases and Non-Gray Absorbing Particles and Non-Gray Scatters in Non-Gray Enclosures[END_REF]:

𝑑𝐼 𝑔 0 𝑑𝑠 + 𝑘 * (𝑔 0 )𝐼 𝑔 0 = 𝑘 * (𝑔 0 )𝑎(𝑔 0 )𝐼 𝑏 (𝑇) (3)
where 𝑔 0 corresponds to the quadrature-point of a 10-point Gauss-Chebyshev quadrature scheme, the index 0 referring to the reference state. The reference state is defined by the mole fractions of CO2 (𝑥 𝐶𝑂2 ), H2O (𝑥 𝐻2𝑂 ), and the SVF (𝑓 𝑆 ) averaged over the flame volume. The reference temperature is calculated as the blackbody emission averaged temperature [START_REF] Modest | Assembly Full Spectrum k-Distribution from a Narrow Band Database: Effects of Mixing Gases, Gases and Non-Gray Absorbing Particles and Non-Gray Scatters in Non-Gray Enclosures[END_REF]. The stretching factor, 𝑎(𝑔 0 ), is defined as 𝑎(𝑔 0 ) = 𝑑𝑔(𝑘 0 ,𝜙 0 ,𝑇) 𝑑𝑔 0 (𝑘 0 ,𝜙 0 ,𝑇 0 ) where 𝜙 = {𝑥 𝐶𝑂2 , 𝑥 𝐻2𝑂 , 𝑓 𝑆 , 𝑇}.

An efficient FS database was developed following the methodology proposed in Ref. [START_REF] Wang | Full-spectrum k-distribution look-up table for non homogeneous gas-soot mixtures[END_REF].

Values of k* and a are stored at the 10 quadrature points for different mixtures of CO2, H2O, and soot. At each grid point and for each quadrature point, the values of k* and a are extracted from the database using a linear interpolation on 𝑥 𝐶𝑂2 , 𝑥 𝐻2𝑂 , and 𝑓 𝑆 , and spline interpolations on 𝑇 and 𝑇 0 .

The RTE is solved by the Finite Volume Method [START_REF] Chui | Prediction of Radiative Transfer in Cylindrical Enclosures with the Finite Volume Method[END_REF] with an angular mesh with 12×16 control angles.

Computational details

Simulations are considered with an overall computational domain of 4cm (r) × 6cm (z). The computational domain is divided into 167 (r) × 378 (z) cells using a non-uniform grid. The finest resolution (76m×76m) is located in a region covering the molten droplet and the sooting region.

Results and discussions

Data analysis

The experimental data are analyzed to determine n in Eq. ( 1). The analysis assumes that the flame spreads in an opposed flow of free stream velocity, 𝑢 ∞ , over a thermally-thin cylindrical material. The surface temperature of the material, 𝑇 𝑃 , remains constant during the pyrolysis process and the gas phase conductivity, 𝜆 𝑓𝑙 , dynamic viscosity, 𝜇 𝑓𝑙 , and flame temperature, 𝑇 𝑓𝑙 , are independent of pressure. A unit Schmidt number is assumed. The gaseous fuel, 𝐹, reacts with the oxidizer, 𝑂 𝑥 , according to the one-step reaction 𝐹 + 𝑆𝑂 𝑥 → (1 + 𝑆)𝑃, where 𝑆 is the oxidizer-fuel mass ratio.

The experimental data reported in Fig. 2a shows that 𝑢 𝑝 and, consequently, 𝑚̇𝑝 are independent of pressure. A scaling analysis, similar to that presented in Ref. [START_REF] Contreras | Oxygen index effect on the structure of a laminar boundary layer diffusion flame in a reduced gravity environment[END_REF], shows that the flame length, 𝐿 𝑓𝑙 varies as

(1+𝑆)𝑚̇𝑝 𝜇 𝑓𝑙
and, then, is independent of P whereas the flame radius, 𝛿 𝑓𝑙 varies as

√ (1+𝑆)𝑚̇𝑝 𝜌 ∞ 𝑢 ∞
and, then, scales with 𝑃 -1/2 .

In the present study, the characteristic length scale for soot formation, 𝐿 𝑆 , is defined as the distance from the molten droplet leading edge, where pyrolysis is supposed to first occur, to the peak of integrated SVF as prescribed in [START_REF] Contreras | Oxygen index effect on the structure of a laminar boundary layer diffusion flame in a reduced gravity environment[END_REF]. Figure 2a shows that, over the range from 70.9 kPa to 121.6 kPa, 𝐿 𝑆 is independent of pressure. The value of 𝐿 𝑆 at 50.7 kPa is significantly lower than those at higher pressure. However, it should be pointed out that the measured SVF at P=50.7 kPa becomes close to the limit of detection of the BMAE technique. These experimental data support that the SF residence time, defined as 𝑡 𝑟𝑒𝑠 = 𝐿 𝑆 𝑢 ∞ ⁄ , is also independent of pressure. Consequently, the soot particles experience the same residence time in the SF region for all the flames investigated in this study.

The conservation of SVF in a Lagrangian sense in the SF region is expressed as the SVF peak, 𝑓 𝑠,𝑚𝑎𝑥 , scales as 𝑃 𝑛 since 𝜔̇𝑠 𝑓,𝑃 scales as 𝑃 𝑛 and both the soot density, 𝜌 𝑠 , and 𝑡 𝑟𝑒𝑠 are independent of pressure. Figure 2b shows the measured 𝑓 𝑠,𝑚𝑎𝑥 as a function of 𝑃/𝑃 0 . 𝑛 is found to be 2.92, showing that the SF rate in the present flames is approximately third-order in pressure. This value of 𝑛 is used in the numerical simulations (see Eq.( 1)).

Model validation

Figure 2b shows that predicted SVF peaks are in proper agreement with the measurements. The largest discrepancies are observed for P=50.7 kPa with the experimental peak being underpredicted by about 36%. More detailed comparisons are presented in Fig. 3 that displays radial profiles of SVF at different locations along the wire axis. Figure 1 shows that the selected where the SVF peaks, in the soot oxidation region, and in the region where soot is fully oxidized for the non-smoking flames, respectively. Section 4.1 revealed that, whatever the pressure, soot particles experience the same residence times at a given location along the wire axis. As a consequence, the observed increase in SVF with pressure at a given 𝑧 is only attributed to pressure effects. Model predictions exhibit an overall reasonable agreement with the experimental trends in each region of the flame. For a given pressure, predicted SVF increases up to 11 mm and then decreases as the distance is further increased in agreement with experimental data. Figure 3d shows that soot is observed at z = 17 mm only for pressure higher than 95 kPa. Therefore, the model predicts correctly that flames at 𝑃 <95kPa are non-smoking whereas those at 𝑃 ≥95kPa are smoking (see Fig. 3d). For a given location, the increase in soot volume fraction with pressure is also well captured and the shape of the profiles as well as the peak values are in reasonable agreement with the experimental data. However, the computed maximum occurs at higher r and the soot region is shifted downstream as compared to the experiments. These discrepancies become more and more pronounced as the distance along the wire axis increases.

Effects of pressure on radiant fraction

Model results show that, for heat transfer purposes, the flames investigated in the present study are optically-thin. Nevertheless, it should be pointed out that the optically-thin assumption may not be generalized at lower velocity since, as shown by Bedir and T'ien [START_REF] Bedir | A computational study of flame radiation in PMMA diffusion flames including fuel vapour participation[END_REF], flames thickens at low stretch rates and self-absorption may become non-negligible.

𝜒 𝑅 = 𝐶𝜅 𝑝 𝑃 -1 (4)
where C does not depend on pressure. Figure 4a displays the evolution of 𝜒 𝑅 as a function of 𝑃. The radiative loss results from the contributions of both radiating gas, 𝜒 𝑅,𝑔 and soot, 𝜒 𝑅,𝑠 .

The radiative loss is dominated by gas radiation (𝜒 𝑅,𝑔 > 𝜒 𝑅,𝑠 ) for pressure lower than about 75 kPa whereas the soot contribution prevails for higher pressure (𝜒 𝑅,𝑠 > 𝜒 𝑅,𝑔 ). 𝜅 𝑝 is proportional to 𝑃 for radiating gas which explains from Eq. ( 4) that 𝜒 𝑅,𝑔 remains constant over the range of pressure investigated (see Fig. 4a). 𝜅 𝑝 varies with pressure as 𝑓 𝑠,𝑚𝑎𝑥 for soot and, then, scales with 𝑃 2.92 according to Section 4.1. This explains from Eq. ( 4) that 𝜒 𝑅,𝑠 scales approximately with 𝑃 1.92 (see Fig. 4a). Therefore, the observed increase in 𝜒 𝑅 with P, from 0.17 at 𝑃 = 50.7 kPa to 0.35 at 𝑃 = 121.6 kPa, results specifically from an increase in the soot contribution (see Fig. 4a).

Figure 4a shows that 𝜒 𝑅 at the SP transition (𝑃 ≈ 95𝑘𝑃𝑎) is close to 0.3 that is consistent with the values reported for axisymmetric laminar diffusion flames at earth gravity [START_REF] Markstein | Relationship between smoke point and radiant emission from buoyant turbulent and laminar diffusion flames[END_REF], but lower than those reported in microgravity [START_REF] Urban | Structure and soot properties of nonbuoyant ethylene/air laminar jet diffusion flames[END_REF]. The SP mechanism is intimately related to the increase in radiative cooling with P. At the SP transition, the temperature at the flame tip becomes too low to enable complete soot oxidation and soot is then released from the flame. In previous studies at earth gravity, the soot oxidation freezing temperature, 𝑇 𝑠 , was defined at the height of smoking flames where the integrated SVF starts to reach a plateau [START_REF] Kent | Temperature and fuel effects in sooting diffusion flames[END_REF]. This definition is adopted in the present study and both numerical and experimental data are reported in Fig. 4b. Both computed and experimental soot burnout temperature stand in the range 1300-1450 K which is also consistent with recent [START_REF] Bonnety | Probing the local radiative quenching during the transition from a non-smoking to a smoking laminar coflow ethylene/air non-premixed flame[END_REF] and older measurements [START_REF] Markstein | Radiant emission and absorption by ethylene and propylene diffusion flames[END_REF], [START_REF] Kent | Why do diffusion flames emit smoke?[END_REF], [START_REF] Kent | Temperature and fuel effects in sooting diffusion flames[END_REF] at earth gravity. 𝑟 𝑤𝑖𝑟𝑒 , is also plotted (left y-axis).

Effects of pressure on the radiative feedback to the solid surface

Conclusions

The effects of pressure on soot production and radiative heat transfer in the non-buoyant opposed-flow flame spread along NiCr wires coated with LDPE are investigated experimentally and numerically. The following conclusions can be drawn:

1) Experimental data show that spread rate, pyrolysis rate, and residence time for soot formation are independent of pressure whereas the soot formation rate is third-order in pressure. r wire q' ' R,w ; P =101.3 kPa q' ' R,inc ; P =50.7 kPa q' ' R,inc ; P =70.9 kPa q' ' R,inc ; P =90.0 kPa q' ' R,inc ; P =101.3 kPa q' ' R,inc ; P =121.6 kPa z b,max

3) The radiant fraction increases with pressure specifically as a result of an increase in soot radiation. In addition, gas radiation dominates soot radiation at pressure lower than 75kPa whereas the opposite trend is observed for higher pressure.

4) The radiant fraction at the SP transition, which occurs at 95kPa, approximately, is about 0.3 and the soot oxidation freezing temperature is found to be in the range 1350-1450 K. These data are consistent with those previously reported at earth gravity. 5) Whatever the pressure considered, the surface re-radiation dominates the incident radiative flux from the flame along the entire wire.

  Key Words: Opposed-Flow Flame Spread, Electrical Wire, Microgravity, Pressure Effects, Soot Radiation Nomenclature a stretch function [-] AS soot surface area [m -1 ]f k-distribution function [m -1 ] fs soot volume fraction [-] g cumulative k-distribution function [-] 𝑔 0 cumulative k-distribution function in the reference state[-] 𝐼 𝑔 , 𝐼 radiative intensity [W•m -2 •sr -1 ] Ib blackbody intensity (Planck function) [W•m -2 •sr -1 ] k absorption coefficient variable [m -1 ] 𝐿 𝑓𝑙 flame length [m]𝐿 𝑆 distance from the molten droplet leading edge to the integrated SVF peak [m] lsp laminar smoke point height [m] 𝑚̇𝑝 pyrolysis mass flow rate [kg•s -1 ] 𝑚̇𝑝 ′′ pyrolysis mass flow rate per unit area [kg•m -2 •s -1 ] NA Avogadro number [part•mol -1 ] NCmin number of carbon atoms in the incipient soot particle [-] NS soot number density per unit mass of mixture [part•kg -1 ] Wi molecular weight of the i th species [kg•mol -1 ] xi mole fraction of the i th species [-] z axial coordinate [m]  stand-off distance [m] 𝛿 𝑓𝑙 flame radius [m]

Figure 1

 1 Figure1shows the experimental fields of SVF at different pressure levels. As expected, SVF

Figure 1 .

 1 Figure 1. Fields of experimental SVF (in ppm) as a function of pressure 𝑃. The horizontal

  . This expression can be rewritten as 𝐷(𝜌 𝑠 𝑓 𝑠 ) 𝐷𝑡 = 𝜌 𝑠 𝑓 𝑠,𝑚𝑎𝑥 𝑡 𝑟𝑒𝑠 = 𝜔̇𝑠 𝑓,𝑃 . As a consequence,

Figure 2 .

 2 Figure 2. a) Spread rate (red square), pyrolysis mass flow rate (blue delta) and 𝐿 𝑆 (green circle) as a function of pressure. b) Peak soot volume fraction as a function of 𝑃 𝑃 0 ⁄ .

  at 𝑧 = 8 mm, 11 mm, 14 mm, and 17 mm are in the soot growth region, in the region

Figure 3 .

 3 Figure 3. Radial profile of SVF at different locations along the wire axis, i.e. a) 8mm, b)

Figure 5

 5 Figure 5 shows the incident radiative flux on the wire surface, 𝑞̇𝑅 ,𝑖𝑛𝑐 ′′ , as a function of the

Figure 4 .

 4 Figure 4. a) Radiant fraction and b) soot quenching temperature as a function of 𝑃. In Fig.4a,

Figure 5

 5 Figure5shows that 𝑞̇𝑅 ,𝑖𝑛𝑐 ′′

Figure 5 Figure 5 .

 55 Figure5shows that 𝑞̇𝑅 ,𝑤 ′′ is higher than 𝑞̇𝑅 ,𝑖𝑛𝑐 ′′

2 )

 2 A two-equation SP-based soot model is proposed and implemented in a CFD model including detailed chemistry and state-of-the-art radiation model. Model predictions are in satisfactory agreement with the available experimental data and capture the effects of pressure on soot production as well as the transition from a non-smoking to a smoking flame.

Frame 003  29

 29

  𝑉 𝑓𝑙 , is proportional to 𝐿 𝑓𝑙 𝛿 𝑓𝑙 2 and, according to Section 4.1, scales with 𝑃 -1 . It is expected that 𝜒 𝑅 depends on pressure through 𝜅 𝑝 𝑉 𝑓𝑙 and then, introducing the pressure dependence of 𝑉 𝑓𝑙 , through 𝜅 𝑝 𝑃 -1 :

			The radiant fraction can
	then be expressed as 𝜒 𝑅 =	4𝜅 𝑝 𝜎𝑇 𝑓𝑙 4 𝑉 𝑓𝑙 𝑚̇𝑝Δℎ 𝑐	. The flame volume,

Acknowledgments

The authors feel grateful to the Centre National d'Etudes Spatiales for its financial support under Contract No. 130615.


Yuying Liu

Jinyu Zhu

Guanghai Liu
email: ghliu@buaa.edu.cn

Jean-Louis Consalvi

Fengshan Liu

A Nongray-Wall Emissivity Model for the Wide-Band Correlated K-Distribution Method

Keywords: Radiative heat transfer, Wide-Band Correlated K-distribution, Emissivity, Nongray wall

The walls of combustion systems are usually assumed to be black or gray in radiative calculations, which may introduce large errors. The Planck-function-weighted emissivity is usually used as the gray-wall emissivity when using the Wide-Band Correlated K-distribution (WBCK) method. This approach can demonstrate good accuracy when the emissivity-based optimized band interval proposed by Solovjov (2013, JQSRT)(WBCK-1) is used. To improve efficiency without losing accuracy, a nongray-wall emissivity model is proposed. The accuracy of this model is first demonstrated for the emissivitybased band interval (WBCK-2). An absorption-coefficient-based optimized band interval is then proposed and the accuracy of this band interval coupled with the nongray-wall emissivity model (WBCK-3) is evaluated. The performance of these models is evaluated in three 1D isothermal and homogeneous cases bounded by fly-ash deposit, GH536, and soot deposit and a 3D fuel-air flame bounded by fly-ash deposit. The results show that WBCK-2 is more accurate than WBCK-1 when the number of bands is greater than 1. WBCK-3 can become more accurate than WBCK-1 and WBCK-2 when the number of bands is larger than 2, especially for low wall temperatures. Furthermore, it is sufficient to divide the spectrum into 3 bands for accurate predictions of radiative calculations.

Introduction

Radiative heat transfer plays an important role in high-temperature industrial combustion systems, such as coal-fired boilers and gas turbine combustors [START_REF] Modest | Radiative heat transfer in turbulent combustion systems: theory and application[END_REF]. Most studies assume that the walls of combustors are black or gray in radiative transfer calculations for simplicity in spite of the fact that the wall materials have different spectral emissivity. This simplification may introduce large errors in the prediction of radiative source term and heat flux [START_REF] Wall | The character of ash deposits and the thermal performance of furnaces[END_REF], which may strongly affect the predicted performance of the combustion devices in terms of thermal protection [START_REF] Dannecker | Impact of radiation on the wall heat load at a test bench gas turbine combustion chamber: measurements and CFD simulation[END_REF] and pollutant emissions [START_REF] Ren | Monte carlo simulation for radiative transfer in a high-pressure industrial gas turbine combustion chamber[END_REF]. Therefore, comprehensive and accurate radiative models are necessary for accurately predicting radiative heat transfer in practical problems involving nongray walls and for optimal design of combustion devices.

Radiative heat transfer models with different spectral resolutions, such as the Line-By-Line method (LBL), narrow-band models [START_REF] Goody | A statistical model for water-vapor absorption[END_REF]-[], wide-band models [START_REF] Ströhle | On the application of the exponential wide band model to the calculation of radiative heat transfer in one-and two-dimensional enclosures[END_REF], [] and full-spectrum models [START_REF] Modest | Narrow-band and full-spectrum k-distributions for radiative heat transfercorrelated-k vs. scaling approximation[END_REF], 11]-13], provide different accuracy and computational efficiency [START_REF] Modest | Radiative heat transfer[END_REF]. The benchmark LBL method is rarely applied to practical applications due to the huge requirement in computing resources [START_REF] Chu | A comprehensive evaluation of the non gray gas thermal radiation using the lineby-line model in one-and two-dimensional enclosures[END_REF]. Narrow-band models, such as the Statistical Narrow-Band method (SNB) [START_REF] Goody | A statistical model for water-vapor absorption[END_REF], ], the Statistical Narrow-Band Correlated-K method (SNBCK) [START_REF] Lacis | A description of the correlated k distribution method for modeling non-gray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres[END_REF] and the Narrow-Band Correlated K-distribution method (NBCK) [START_REF] Modest | Narrow-band and full-spectrum k-distributions for radiative heat transfercorrelated-k vs. scaling approximation[END_REF], demonstrate the best accuracy in comparison to LBL because of small changes in spectral emissivity within a narrow spectral range for both gases and a solid wall. Although the number of radiative transfer equations (RTEs) of narrow-band models is much less than that of LBL, the computing resources required for solving narrow-band RTEs are still unacceptable in practical applications. Therefore, various computationally more efficient wide-band models and full-spectrum models have been developed. Full-spectrum models, such as the Full-Spectrum Correlated K-distribution method (FSCK) [START_REF] Modest | Narrow-band and full-spectrum k-distributions for radiative heat transfercorrelated-k vs. scaling approximation[END_REF], the Spectral-Linebased Weighted-Sum-of-Gray-Gases method (SLW) [START_REF] Denison | A spectral line-based weighted sum of gray gases model for arbitrary RTE solvers[END_REF] and the Weighted-Sum-of-Gray-Gases method (WSGG) [START_REF] Johansson | Account for variations in the H2O to CO2 molar ratio when modelling gaseous radiative heat transfer with the Weighted-Sum-of-Grey-Gases Model[END_REF], provide the highest computational efficiency at the cost of some accuracy for practical problems involving nongray walls [START_REF] Solovjov | Efficient cumulative wavenumber model of radiative transfer in gaseous media bounded by non-gray walls[END_REF], 17]. Wide-band models have a great potential for achieving the best compromise between efficiency and accuracy for practical problems with nongray walls. For example, Solovjov et al. [START_REF] Solovjov | Efficient cumulative wavenumber model of radiative transfer in gaseous media bounded by non-gray walls[END_REF] proposed a method to optimize band interval according to the spectral emissivity for the expansion of SLW named the cumulative wavenumber model, in which only a few RTEs need to be solved. Fonseca et al. [START_REF] Fonseca | Application of the WSGG model to solve the radiative transfer in gaseous systems with nongray boundaries[END_REF] separated gray gases into several groups according to a presumed emissivity, followed by solving RTE with the WSGG method. The banded method was employed by Bordbar et al. [START_REF] Bordbar | Line by line based band identification for non-gray gas modeling with a banded approach[END_REF], 19] to solve such problems as well. It was shown that those methods provide a good compromise between accuracy and efficiency for nongray wall problems.

The Planck-function-weighted emissivity at the wall temperature is usually used as the gray-wall emissivity to ensure the conservation of radiative energy at the wall [START_REF] Modest | Radiative heat transfer[END_REF]. This gray-wall emissivity model can achieve good accuracy when combined with the emissivity-based optimized band interval mentioned above [START_REF] Solovjov | Efficient cumulative wavenumber model of radiative transfer in gaseous media bounded by non-gray walls[END_REF] for nongray wall problems too. To further improve computational efficiency without significantly losing accuracy, some methods have been proposed recently. For instance, Fonseca et al. [START_REF] Fonseca | A simplified approach to compute the radiative transfer in a participating media bounded by non-gray walls[END_REF] used a radiating medium mean temperature, rather than the wall temperature, as the Planck temperature to obtain the gray-wall emissivity. Liu et al. [START_REF] Liu | A wavenumber subinterval grouping strategy compatible with non-gray metal wall boundaries used in multi-scale multi-group full spectrum k-distribution gas radiation model[END_REF] proposed a grouping strategy to improve the correlation between the absorption spectrum of the radiating medium and the emission spectrum of the nongray wall and then solved RTE with the Multi-Scale Multi-Group FSK method (MSMGFSK). It was shown that these two methods can demonstrate good accuracy and computational efficiency.

The objective of this work is to propose a nongray-wall emissivity model for the Wide-Band Correlated K-distribution (WBCK) method and compare this nongray-wall emissivity model with, on one hand, benchmarks solutions obtained from LBL calculations and, on the other hand, the gray-wall one, i.e., the Planck-function-weighted emissivity, in three 1D isothermal and homogeneous cases and a 3D fuel-air flame. All the 1D test cases are bounded by nongray walls composed of three typical materials found in industrial combustion devices, namely fly-ash deposit, GH536, and soot deposit, while the 3D case is bounded by walls coated with the soot deposit. Similar to the treatments of Solovjov et al. [START_REF] Solovjov | Efficient cumulative wavenumber model of radiative transfer in gaseous media bounded by non-gray walls[END_REF] and Fonseca et al. [START_REF] Fonseca | Application of the WSGG model to solve the radiative transfer in gaseous systems with nongray boundaries[END_REF], the present WBCK method is formulated by generating the Planck-function-weighted wide-band k-distribution. This article is organized as follows. The second section presents the theoretical background. The third and fourth sections are devoted to the comparison of different models in three 1D cases and one 3D fuel-air flame, respectively. Section 5 summarizes the conclusions drawn from the present work.

Theoretical background

WBCK RTE

The spectral RTE in participating and non-scattering media can be written as:

𝑑𝐼 𝜂 𝑑𝑠 = 𝜅 𝜂 (𝜙) [𝐼 𝑏𝜂 (𝑇) -𝐼 𝜂 ] (1)
where 𝜂 is the wavenumber, 𝐼 𝜂 is the spectral radiative intensity along path s, 𝐼 𝑏𝜂 is the spectral Planck function, 𝜅 𝜂 is the spectral absorption coefficient, and 𝜙 is the vector of local state variables consisting of temperature, T, total pressure, P, and mole fraction vector, 𝑥.

Similar to the full-spectrum k-distribution function, the wide-band k-distribution function, 𝑓 𝑚 , which is weighted by the Planck function, is defined as [START_REF] Modest | Narrow-band and full-spectrum k-distributions for radiative heat transfercorrelated-k vs. scaling approximation[END_REF][8]:

𝑓 T 𝑃 ,𝜙,𝑚 (𝑘 𝑚 ) = 1 𝐼 𝑏,𝑚 (T 𝑃 ) ∫ 𝐼 𝑏𝜂 (T 𝑃 )𝛿 (𝑘 𝑚 -𝜅 𝜂 (𝜙)) d𝜂 𝜂 𝑢,𝑚 𝜂 𝑙,𝑚 (2)
where (3) Thus, the cumulative function 𝑔 𝑚 is a monotonically increasing function varying between 0 and 1.

The RTE in the wavenumber-space can be converted into the reordered form in the reference g0-space by multiplying with 𝛿(𝑘 𝑚 -𝜅 𝜂 (𝜙 0 ))

𝑓 𝑇 0 ,𝜙 0 ,𝑚 (𝑘 𝑚 ) , followed by integrating over all the wavenumbers belonging to the mth band:

𝑑𝐼 𝑔 0 ,𝑚 𝑑𝑠 = 𝑘 𝑚 * (𝑔 0 )[𝑎 𝑚 (𝑔 0 )𝐼 𝑏,𝑚 (𝑇) -𝐼 𝑔 0 ,𝑚 ] (4)
where the subscript 0 refers to the reference state. In Eq. ( 4), 𝐼 𝑔 0 and the non-gray stretching factor, 𝑎 𝑚 (𝑔 0 ), are expressed as:

𝐼 𝑔 0 ,𝑚 = ∫ 𝐼 𝜂 𝛿(𝑘 𝑚 -𝜅 𝜂 (𝜙 0 ))𝑑𝜂 𝜂 𝑢,𝑚 𝜂 𝑙,𝑚 𝑓 𝑇 0 ,𝜙 0 ,𝑚 (𝑘 𝑚 ) (5)
𝑎 𝑚 (𝑔 0 ) = 𝑓 𝑇,𝜙 0 ,𝑚 (𝑘 𝑚 )

𝑓 𝑇 0 ,𝜙 0 ,𝑚 (𝑘 𝑚 ) = 𝑑𝑔 𝑇,𝜙 0 ,𝑚 (𝑘 𝑚 )

𝑑𝑔 𝑇 0 ,𝜙 0 ,𝑚 (𝑘 𝑚 ) (6)
The conventional correlated reordered absorption coefficient scheme proposed by Modest et al. [START_REF] Modest | Narrow-band and full-spectrum k-distributions for radiative heat transfercorrelated-k vs. scaling approximation[END_REF] is used here to obtain the correlated reordered absorption coefficient 𝑘 𝑚 * using the following explicit equation:

𝑔 𝑇 0 ,𝜙,𝑚 (𝑘 𝑚 * ) = 𝑔 0 (7)
Because the reordered absorption coefficient, 𝑘 𝑚 * , is a monotonically increasing function of 𝑔 0 , the total radiative intensity, I, can be obtained using a N-point quadrature scheme: 𝐼 = ∑ ∑ 𝑤 𝑛 𝐼 𝑔 0 ,𝑚,𝑛 𝑁 𝑛=1 𝑀 𝑚=1 [START_REF] Modest | Narrow-band and full-spectrum k-distributions for radiative heat transfercorrelated-k vs. scaling approximation[END_REF] where M and N are the number of bands and quadrature points, respectively, and 𝑤 𝑛 is the weight of the nth quadrature point. It should be pointed out that the WBCK method degenerates to FSCK when M = 1.

WBCK boundary conditions

In this study, it is assumed that the walls reflect radiation diffusely and such an assumption has been used in many numerical studies [START_REF] Zhao | A transported probability density function/photon Monte Carlo method for high-temperature oxy-natural gas combustion with spectral gas and wall radiation[END_REF], []. Consequently, the boundary condition of RTE in the wavenumber-space (Eq. ( 1)) is given as

𝐼 𝜂,𝑤 = 𝜀 𝜂 𝐼 𝑏𝜂 (𝑇 𝑤 ) ⏟ emission term + (1 -𝜀 𝜂 ) 1 𝜋 ∫ 𝑛 ⃗ •𝑠 <0 𝐼 𝜂 |𝑛 ⃗ • 𝑠 |𝑑Ω ⏟ reflection term (9)
where 𝜀 𝜂 is the spectral emissivity of the wall, 𝑛 ⃗ is the unit vector normal to the wall and towards the medium, and the subscript w refers to the wall.

Gray-wall emissivity model

In most previous studies, the walls are assumed to be black or gray in radiative calculations [START_REF] Zhao | A transported probability density function/photon Monte Carlo method for high-temperature oxy-natural gas combustion with spectral gas and wall radiation[END_REF], ], i.e., the wavenumber dependence of wall emissivity is completely neglected. Similar to the derivation of Eq. ( 4), the boundary condition in the reference 𝑔 0 -space can be given by: 𝐼 𝑔 0 ,𝑚,𝑤 = 𝜀 𝑚 𝑎 𝑚,𝑤 (𝑔 0 )𝐼 𝑏,𝑚 (𝑇 𝑤 ) ⏟ emission term [START_REF] Ströhle | Assessment of the re-ordered wide band model for non-grey radiative transfer calculations in 3D enclosures[END_REF] The Planck-function-weighted emissivity at the wall temperature is used as the gray-wall emissivity, which is expressed as [START_REF] Modest | Radiative heat transfer[END_REF] 𝜀 𝑚 =

+ (1 -𝜀 𝑚 ) 1 𝜋 ∫ 𝑛 ⃗ •𝑠 <0 𝐼 𝑔 0 ,𝑚 |𝑛 ⃗ • 𝑠 |𝑑𝛺 ⏟ reflection term

Nongray-wall emissivity model

In practical applications, the emissivity of walls normally varies with wavenumber. In this case, the emission term in Eq. ( 9) can be integrated over a wide band as

where 𝜀 𝑚 * (𝑔 0 ) is expressed as:

𝜀 𝑚 * (𝑔 0 ) = 1 𝑓 𝑇 𝑤 ,𝜙 0 ,𝑚 (𝑘 𝑚 ) • 1 𝐼 𝑏,𝑚 (𝑇 𝑤 ) • ∫ 𝜀 𝜂 𝐼 𝑏𝜂 (𝑇 𝑤 )𝛿 (𝑘 𝑚 -𝜅 𝜂 (𝜙 0 )) 𝜂 𝑢,𝑚 𝜂 𝑙,𝑚 𝑑𝜂 (13)
Therefore, the boundary condition in the reference 𝑔 0 -space can be rewritten as

𝐼 𝑔 0 ,𝑚,𝑤 = 𝜀 𝑚 * (𝑔 0 )𝑎 𝑚,𝑤 (𝑔 0 )𝐼 𝑏,𝑚 (𝑇 𝑤 ) + (1 -𝜀 𝑚 * (𝑔 0 )) 1 𝜋 ∫ 𝑛 ⃗ •𝑠 <0 𝐼 𝑔 0 ,𝑚 |𝑛 ⃗ • 𝑠 |𝑑𝛺 (14)
It is easy to anticipate that it is important to account for the nongray-wall emissivity when the emission term of Eq. ( 9) is much larger than the reflection term. However, for most combustion devices, the reflection term can also be important. Therefore, the importance of considering the nongray-wall emissivity needs to be evaluated under different wall and medium thermal conditions.

Methods for determining the band interval of WBCK 2.3.1 Emissivity-based method

The optimized band interval method proposed by Solovjov et al. [START_REF] Solovjov | Efficient cumulative wavenumber model of radiative transfer in gaseous media bounded by non-gray walls[END_REF] is obtained through minimization of the following objective function

∑ ∫ |𝜀 𝜂 -𝜀 𝑚 | 𝜂 𝑢,𝑚 𝜂 𝑙,𝑚 𝑀 𝑚=1 𝐼 𝑏𝜂 (T 𝑤 )𝑑𝜂 → min (15)
where the band emissivity, 𝜀 𝑚 , is calculated by Eq. [START_REF] Denison | A spectral line-based weighted sum of gray gases model for arbitrary RTE solvers[END_REF]. In this study, 100 cm -1 is chosen as the band variation step to obtain the optimized band interval for simplicity.

Absorption-coefficient-based method

The selection of emissivity-based method (Eq. ( 15)) may become inaccurate for low temperature walls. Taking the virtual spectral absorption coefficients and spectral emissivity distribution along the wavenumber shown in Fig. 1 as an example, supposing that there are two identical peak spectral absorption coefficients over the entire spectrum locating at 2500 cm -1 and 7500 cm -1 , and the spectral emissivity increases linearly from 0 to 1 for the wavenumber of 0 -10000 cm -1 . It can be seen that the spectral emissivity at the peaks of spectral absorption coefficient are 0.25 and 0.75, as shown in the left frame of Fig. 1(a). The emissivity-based method (Eq. ( 15)) for a wall temperature of 500 K decomposes the spectrum into two bands with the spectral ranges of 0 -1400 cm -1 and 1400 -10000 cm -1 . The right frame of Fig. 1(a) shows that the correlated reordered absorption coefficient in the g0 space, 𝑘 𝑚 * , of the 2nd band is 0 when g0 < 1, and is equal to the value of peak absorption coefficient in the left frame of Fig. 1(a) for g0 = 1. The nongray-wall emissivity in the g0 space, 𝜀 𝑚 * , of the 2nd band is 0 when g0 < 1, and is equal to about 0.25 for g0 = 1 according to Eq. ( 13) due to the Planck function weight. This shows clearly that the emissivity-based method introduces large errors for prediction of emissivity at the large absorption coefficients.

To improve the accuracy of the nongray-wall emissivity for large absorption coefficients that correspond to large reference g0, an absorption-coefficient-based method is proposed here. The main idea of the proposed method is to put large absorption coefficients at different wavenumbers into different wide bands, as show in Fig. 1(b). If the two peak absorption coefficients are divided into two different bands, typically 0 -5000 cm -1 and 5000 -10000 cm -1 , then 𝜀 𝑚 * of the 1st and 2st band is equal to 0.25 and 0.75 when g0 = 1 respectively, which is much more accurate than that of emissivity-based method. For practical radiating medium, the spectral absorption coefficients of CO2 are mainly concentrated in 4 spectral positions, the proposed band interval determination method may provide good accuracy. The optimization procedure is illustrated in Fig. 2, which is for 10% CO2 (the left frame) and 10% H2O (the right frame) at 2500 K and 1 atm as examples. In these two examples, it is assumed that the wall surface emissivity increases linearly between 0 ~ 1 at a wavenumber of 0 ~ 10000 cm -1 . The entire spectrum is divided into 4 bands.

Firstly, the narrow-band mean absorption coefficients over spectral bands of 200 cm -1 is calculated by Eq. ( 16) and the four maximum absorption coefficients, 𝐴 𝑚 , is marked in order in its magnitude.

𝜅 𝑚 ′ = 1 𝜂 𝑢,𝑚 -𝜂 𝑙,𝑚 ∫ 𝜅 𝜂 d𝜂 𝜂 𝑢,𝑚 𝜂 𝑙,𝑚 (16)
Secondly, the wavenumber delimiters of all wide bands, 𝑃 𝑚 , are determined as the spectral locations of the minimum absorption coefficient between the adjacent maximum absorption coefficients, 𝐵 𝑚 , as show in Fig. 2(a) and Fig. 2(b), which takes the third band with an absorption coefficient greater than 0.1 cm -1 (marked by blue dashed line and related vertical pink lines) as an example. It can be seen that the wall emissivity in the wavenumber-space lies in a very small range of 0.34 ~ 0.37 for Fig. 2(a) or 0.33 ~ 0.37 for Fig. 2(b). The relatively high concentration of large absorption coefficients in each wide band makes the associated spectral wall emissivity vary weakly, which results in the reordered wall emissivity is very close to the true value in wavenumber space at larger reference g0 positions.

The reason for choosing the spectral bands of 200 cm -1 as narrow band rather than a narrower one, such as 100 cm -1 , is that the profile of the narrow-band mean absorption coefficients over 200 cm -1 is smoother than over 100 cm -1 , which helps locate the 𝐴 𝑚 and 𝐵 𝑚 . However, it should be pointed out that the choice of narrow-band width in Eq. ( 16) is not critical since it only slightly affects the wavenumber between two adjacent wide bands.

Results and discussion of one-dimensional cases

It is natural to expect that different boundary conditions have different impacts on the predicted radiative source term and heat flux. Here three typical combustor surface materials with different spectral emissivities are considered to compare the accuracy of WBCK-1, WBCK-2 and WBCK-3, including GH536 at 1000 K [START_REF] Kong | Normal spectral emissivity of GH536 (HastelloyX) in three surface conditions[END_REF], fly-ash deposit at 1093 K [START_REF] Markham | Measurement of radiative properties of ash and slag by FT-IR emission and reflection spectroscopy[END_REF], and soot deposit with a thickness of 1 μm [START_REF] Liebert | Spectral emittance of soot[END_REF] (hereafter named as BC-1, BC-2 and BC-3, respectively). Fly-ash deposit, which is typically found in coal-fired boilers, is a non-metallic material. Its spectral emissivity decreases with increasing the wavenumber, as shown in Fig. 3. GH536 [START_REF] Kong | Normal spectral emissivity of GH536 (HastelloyX) in three surface conditions[END_REF], as one of high-temperature alloys, is usually applied to gas turbine combustion chambers. Its spectral emissivity increases smoothly with increasing the wavenumber, Fig. 3. In hydrocarbon fuel combustion systems, soot deposit is often seen on the walls. Its spectral emissivity increases as the wavenumber increases as well, Fig. 3. The accuracy of WBCK-1, WBCK-2 and WBCK-3 with one (i.e., FSCK), two, and four bands is investigated, and the influence of the number of bands will also be discussed later. Fig. 3 The spectral emissivity of fly-ash deposit at 1093 K [START_REF] Markham | Measurement of radiative properties of ash and slag by FT-IR emission and reflection spectroscopy[END_REF], GH536 at 1000 K [START_REF] Kong | Normal spectral emissivity of GH536 (HastelloyX) in three surface conditions[END_REF], and soot deposit with a thickness of 1 μm [START_REF] Liebert | Spectral emittance of soot[END_REF].

In this section, three one-dimensional isothermal and homogeneous cases are investigated, which includes a pure CO2 case, a pure H2O case, and a CO2-H2O mixture case, as shown in Table 1. The discrete-ordinates method (DOM) along with the S8 angular discretization scheme [START_REF] Modest | Radiative heat transfer[END_REF] is used to solve RTE. The spectral absorption coefficients of radiating gases are derived from the HITEMP 2010 database [START_REF] Rothman | HITEMP, the high-temperature molecular spectroscopic database[END_REF] and the HAPI code [START_REF] Kochanov | HITRAN application programming interface (HAPI): a comprehensive approach to working with spectroscopic data[END_REF]. The range of wavenumber covers from 200 𝑐𝑚 -1 to 10000 𝑐𝑚 -1 and the wavenumber interval is set to 0.02 𝑐𝑚 -1 in LBL calculations, which has been proven to be a good compromise between accuracy and efficiency [START_REF] Chu | A comprehensive evaluation of the non gray gas thermal radiation using the lineby-line model in one-and two-dimensional enclosures[END_REF]. The 32-point Gauss-Chebyshev scheme is selected to reduce the error induced by quadrature scheme. For both the gray and non-gray boundary conditions, iteration is necessary in solving the RTEs due to reflection of incident radiative heat flux at walls. The convergence condition for all the bands is 𝑚𝑎𝑥 (

𝐺 𝑚 𝑄 -𝐺 𝑚 𝑄-1 𝐺 𝑚 𝑄 ) < 10 -6 ( 17
)
where Q is the number of iterations and Gm is incident radiation of the mth band evaluated as 𝐺 𝑚 𝑄 = ∑ ∑ 𝑤 𝑛 𝑢 𝑗 𝐼 𝑔 0 ,𝑚,𝑛,𝑗 𝐽 𝑗=1 𝑁 𝑛=1 [START_REF] Bordbar | Line by line based band identification for non-gray gas modeling with a banded approach[END_REF] where 𝑢 𝑗 is weight of the jth ordinate and J is the number of angular ordinates. Table 1. Description of the thermal conditions in the three one-dimensional cases.

Case Number of subcases

Length L (m)

Internal temperature (K) Wall temperature (K) 𝑥 𝐶𝑂 2 𝑥 𝐻 2 𝑂 Boundary condition (a) BC-1 1 (b) 1 2500 300 -2000 0.1 0 BC-2 (c) BC-3 (a) BC-1 2 (b) 1 2500 300 -2000 0 0.1 BC-2 (c) BC-3 (a) BC-1 3 (b) 1 2500 300 -2000 0.1 0.2 BC-2 (c) BC-3
Three metrics are used to evaluate the accuracy of the three WBCK models, including the local relative error of radiative source term (LE(x)), the total absolute relative error of radiative source term (TE), and the error of heat flux at the right wall (EF). These quantities are defined as:

LE(x) = -𝛁•𝑞̇𝑅
where L is the length of computational domain.

The pure CO2 case

Firstly, a pure CO2 case (Case 1) is considered, as shown in Table 1. The gaseous medium bounded by the two walls consists of 10% CO2 (balanced by N2) at 2500 K and 1 atm, while the wall temperature is varied from 300 K to 2000 K. Figure 4 shows the variation of TE with the wall temperature. From the top frame of Figure 4, it can be seen that TE of 1-band WBCK-1 decreases firstly as the wall temperature increases. After reaching the minimum at about 1400 K, 1100 K and 1000 K for BC-1, BC-2, and BC-3, respectively, it increases as the wall temperature further increases. In contrast, TE of 1-band WBCK-2 (or WBCK-3) decreases firstly with increasing the wall temperature and then remains very small (using BC-1) or negligible (using BC-2 and BC-3) at sufficiently high wall temperatures.

The values of TE of 2-and 4-band WBCK-1 display different trends for different wall materials. For BC-1 and BC-3, TE of 2-and 4-band WBCK-1 drops sharply with increasing the wall temperature till around 500 K, and then oscillates around a certain low value at higher wall temperatures, as shown in the middle and bottom rows of Figures 4(a) and 4(c). For BC-2, TE of 2-and 4-band WBCK-1 first decreases quickly to a minimum around 800 to 1000 K wall temperature but then increases quickly at a similar rate with increasing the wall temperature. In contrast, the TE values of 2-and 4-band WBCK-2 first decrease quickly with increasing wall temperature till around 500 K and then remain nearly constant close to zero with further increasing the wall temperature for all the three surface materials. In terms of TE, the accuracy of 2-band WBCK-3 is almost the same as 2-band WBCK-2, while the accuracy of 4-band WBCK-3 remains excellent regardless of the wall temperatures. In conclusion, 1-, 2-and 4-band WBCK-2 and WBCK-3 are much more accurate than their WBCK-1 counterparts, 2-band WBCK-2 is as accurate as 2-band WBCK-3, and 4-band WBCK-3 displays better accuracy compared with 4-band WBCK-2, especially at low wall temperatures below about 500 K. Figure 5 shows the variation of EF of different models and surface materials with the wall temperature. For all the three surface materials, EF of 1-band WBCK-1 first decreases in magnitude with increasing temperature till reaching zero but then increases in magnitude with further increasing the wall temperature. On the other hand, EF of 1-band WBCK-2 (or WBCK-3) decreases in magnitude with increasing temperature and then remains very low close to zero at wall temperatures above 1000 K, regardless of the surface materials, as shown in the top row of Figure 5. Similar to the results shown in Figure 4, the EF values of 2-and 4-band WBCK-1 display different trends for different surface materials. For BC-1 and BC-3, the EF values of 2-and 4-band WBCK-1 first decrease in magnitude with increasing wall temperature and then fluctuate around zero at wall temperatures above about 800 K. As shown in the middle and bottom rows of Figure 5(b), the EF values of 2-and 4band WBCK-1 for BC-2 increase nearly monotonically with increasing the wall temperature, similar to 1-band WBCK-1. The performance of 2-and 4-band WBCK-2 and WBCK-3 models evaluated in terms of EF is similar to that observed in TE shown in Figure 4. In summary, WBCK-1 is consistently less accurate than WBCK-2 and WBCK-3, regardless of the number of bands, wall temperature, or the type of wall materials, i.e., it is necessary to consider the spectral behavior of wall emissivity of the three typical wall materials. When the spectrum is divided into 2-bands, the accuracy of WBCK-2 and WBCK-3 is very similar. However, WBCK-3 is very accurate when the spectrum is divided into 4 wide-bands, regardless of the wall temperature or the wall material. Figure 6 shows the distributions of radiative source term for wall temperature of 500 K and boundary condition of BC-3, corresponding to conditions marked by the vertical cyan dashed lines in Figures 4(c To explain why WBCK-2 is more accurate than WBCK-1, the left frame of Fig. 7 displays the 2nd-band spectral absorption coefficient of 2-band WBCK-1, the normalized spectral Planck function, 𝐼 𝑏𝜂 /𝐼 𝑏 (Tw = 500 K), and the spectral emissivity of BC-3 as a function of wavenumber. The right frame of Fig. 7 shows the reordered absorption coefficient, 𝑘 * , and the 2ndband reordered emissivity of WBCK-1 and WBCK-2 as a function of 𝑔 0 . As is known, larger absorption coefficients are in general more important than smaller ones in radiative heat transfer, as shown in WBCK RTE (Eq. ( 4)). The spectral absorption coefficients of CO2 at the 4.3 μm band (i.e., centered at 2410 cm -1 ) are much larger than those at other spectral positions [START_REF] Modest | Radiative heat transfer[END_REF], which means that the spectral emissivity near 2410 cm -1 has a great impact on the predicted radiative heat transfer. The gray horizontal dashed line marks the absorption coefficients larger than 0.05 cm -1 (𝜅 for the wavenumber-space and k* for the g0-space). The cyan horizontal dashed lines mark the corresponding range of spectral emissivity, which is from 0.62 to 0.65 (see the left frame of Figure 7). The emissivity of WBCK-2 for absorption coefficients larger than 0.05 cm -1 is between 0.62 and 0.64, and very close to the spectral emissivity in the wavenumber-space. As for WBCK-1, the emissivity of WBCK-1 is 0.59 due to the influence of spectral Planck function weighting (see the red line of left frame of Figure 7). In other words, the emissivity of WBCK-2 is more accurate than that of WBCK-1, resulting in better accuracy of WBCK-2 than WBCK-1. Furthermore, from Fig. 6(c), it can be found that LE(x) of 4-band WBCK-3 is negligible, while LE(x) of 4-band WBCK-2 remains nearly constant around 1%. This is attributed to the different band intervals of WBCK-2 and WBCK-3. For WBCK-2, the four bands are 200-900 cm -1 , 900-1400 cm -1 , 1400-2000 cm -1 and 2000-10000 cm -1 , while for WBCK-3, the four bands are 200-1300 cm -1 , 1300-2700 cm -1 , 2700-4300 cm -1 and 4300-10000 cm -1 . Figure 8 displays the 4th-band emissivity of 4-band WBCK-2. The gray horizontal dashed lines mark absorption coefficients that lie between 0.001 cm -1 and 0.01 cm -1 , and the cyan horizontal dashed line indicate the spectral emissivity of BC-3 (between 0.75 and 0.78, see the left frame of Figure 8). However, the emissivity predicted by WBCK-2 is around 0.69, and is much smaller than the spectral emissivity in the wavenumber-space. Figure 9 shows the 3rd-band emissivity of 4-band WBCK-3. The emissivity of WBCK-3 ranges from 0.75 to 0.77. In other words, the reordered emissivity predicted by WBCK-3 shows better agreement with the spectral emissivity in the wavenumber-space than it predicted by WBCK-2. This explains why WBCK-3 is more accurate than WBCK-2 at a low wall temperature.

The pure H2O case

The second test case (Case 2) contains pure H2O as the radiating gas, whose thermal conditions are given Table 1. The medium is composed of 10% H2O (balanced by N2) at 2500 K at 1 atm, while the wall temperature is varied from 300 K to 2000 K. Figures 10 and 11 display the variations of TE and EF with the wall temperature. It can be seen that the trends of TE and EF of 1-band WBCK-1 and WBCK-2 (or WBCK-3) are consistent with those observed in Case 1, i.e., TE and EF of 1band WBCK-1 decrease in magnitude firstly as the wall temperature increases. After reaching the minimum, they increase in magnitude with further increasing the wall temperature. In contrast, TE and EF of 1-band WBCK-2 (or WBCK-3) decrease continuously with the increase of the wall temperature. Different from the pure CO2 case, 1-band WBCK-1 shows better accuracy than 1-band WBCK-2 (or WBCK-3) except for BC-2 and BC-3 with a high wall temperature. This may be attributed to the different radiative characteristics of CO2 and H2O. Compared with CO2, which radiates predominately in four spectral regions [START_REF] Modest | Radiative heat transfer[END_REF], H2O radiates almost continuously over the entire spectrum. Overall, the 2-and 4-band WBCK-2 models are more accurate than the corresponding WBCK-1 models, except when BC-3 is the boundary material. For BC-3, the values of both TE and EF of 2-and 4-band WBCK-2 are almost the same as those of WBCK-1 when the wall temperature is lower than 1000 K and 700 K, respectively. The 2-band WBCK-3 model displays lower accuracy than both WBCK-1 and WBCK-2; however, when the entire spectrum is divided into 4 bands, WBCK-3 is more accurate than WBCK-1 and WBCK-2.

The CO2-H2O mixture case

In the final 1D test case (Case 3), the medium is a CO2-H2O mixture and the thermal conditions are also given in Table 1. The gaseous medium consists of 10% CO2 and 20% H2O (balanced by N2) at 2500 K and 1 atm, while the wall temperature is varied from 300 K to 2000 K.

The influence of number of bands

As displayed above, WBCK-3 shows worse accuracy than WBCK-2 when the spectrum is divided into 2 bands; however, it displays significantly better accuracy when the number of bands reaches 4. In this section, the influence of the number of bands is investigated for Case 3 at the wall temperature of 300 K and 1500 K.

Figure 14 and 15 shows the variation of TE with the number of bands. It can be seen that, for BC-1 and BC-3 at wall temperature of 300 K, TE of both WBCK-1 and WBCK-2 decreases with increasing the number of bands, though somewhat gradually, while it displays much weaker dependence on the number of bands for BC-2. TE of WBCK-3 drops rapidly with increasing the number of bands from 1 to 3 for all the three wall materials and then remains close to zero as the number of bands further increases beyond 3.

For the wall temperature of 1500 K, the TE trends of WBCK-1 are the same as it performed in Fig. 14. In contrast, the TE trends of WBCK-2 and WBCK-3 are almost consistent with those of WBCK-3 in Fig. 14,i

Results and discussion of a three-dimensional fuel-air flame

In this section, the accuracy of 1-and 4-band WBCK-1, WBCK-2 and WBCK-3 in a 3D enclosure containing a fuel-air flame [START_REF] Bordbar | Improved banded method for spectral thermal radiation in participating media with spectrally dependent wall emittance[END_REF], [], ] is evaluated. The combustion chamber is a rectangular enclosure of 2 m × 2 m × 4 m. The wall temperature is set to 300 K [START_REF] Liu | Numerical solutions of three-dimensional non-grey gas radiative transfer using the statistical narrow-band model[END_REF], and BC-1 is used as the wall material here, which is different from Refs. [START_REF] Bordbar | Improved banded method for spectral thermal radiation in participating media with spectrally dependent wall emittance[END_REF], [], ]. A mixture of 10% CO2, 20% H2O and 70% N2 is uniformly distributed in the combustion chamber. The temperature profile within the combustion chamber is axisymmetric, which is given by Equations ( 22) and ( 23) 𝑇 = (𝑇 𝑐 -𝑇 𝑒 )𝑓(𝑟 𝑐 /𝑅) + 𝑇 𝑒 (22) 𝑓(𝑟 𝑐 /𝑅) = 1 -3(𝑟 𝑐 /𝑅) 2 + 2(𝑟 𝑐 /𝑅) 3 (23) where 𝑇 𝑐 is the temperature along the centerline (0, 0, z), 𝑇 𝑒 is the temperature at the exit of combustion chamber (z = 4 m), 𝑅 is the radius of 1 m. The temperature along the centerline of combustion chamber increases linearly from 400 K at z = 0 to 1800 K at z = 0.375 m, then it decreases linearly to the 800 K at z = 4 m. The gas temperature in the corner regions, where rc > R, is set to the exit temperature of Te = 800 K.

The number of computational grids is 17 × 17 × 24. The grid is uniformly distributed in the x and y directions, and nonuniformly distributed in the z (streamline) direction. The details of grid are consistent with those used in Liu [START_REF] Liu | Numerical solutions of three-dimensional non-grey gas radiative transfer using the statistical narrow-band model[END_REF]. DOM with the T4 [START_REF] Thurgood | The TN quadrature set for the discrete-ordinates method[END_REF] angular discretization scheme is used to solve RTE. NBCK with band interval of 50 cm -1 is used as the benchmark because the computational resources required by LBL in this 3D case are unacceptably large The 8-point and 16-point Gauss-Chebyshev quadrature schemes are used for the NBCK and WBCK methods, respectively. The emission-weighted temperature is chosen as the reference temperature when using the WBCK method [START_REF] Modest | The full-spectrum correlated-k distribution for thermal radiation from molecular gas-particulate mixtures[END_REF].

Figure 16 displays the radiative source terms along the enclosure centerline (0, 0, z) predicted by NBCK (bounded by black walls and BC-1), 1-band WBCK-1, 1-band WBCK-2, and 4-band WBCK-1, WBCK-2 and WBCK-3. It can be found that the wall material (black or non-black) has some effects on the radiative source term near the wall, as shown in Figure 16(a) (compare the results of NBCK with the black walls and BC-1). The results of 1-band WBCK-1 and WBCK-2 are almost the same, as already observed in Figure 12, and the maximum LE(x) in magnitude is about -8.7% and -8.8%. The predictions of 4-band WBCK-1 and WBCK-2 are also almost the same, Fig. 16(b), which is also similar to the results shown in Fig. 12. When using 4 bands, WBCK-3 can effectively improve the accuracy of WBCK-1 and WBCK-2. The maximum LE(x) in magnitude of WBCK-1, WBCK-2 and WBCK-3 are -8.8%, -9.0% and -1.2%, respectively, as shown in the top frame of Fig. There are obvious discrepancies between the NBCK results for the two boundary conditions (black wall vs. BC-1). For the black walls, the maximum flux along the line of (1 m, 1 m, z) is 20.5 kw/m 2 , which is consistent with the result of Porter et al. [START_REF] Porter | Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments[END_REF], while for BC-1, the maximum net flux is lower at 16.1 kw/m 2 . Similar to the radiative source term along the center line (0, 0, z), 1-band WBCK-1 and WBCK-2 predicted undistinguishable results but with large errors. When using 4 bands, WBCK-1 and WBCK-2 yield significantly better accuracy than their 1-band counterparts and their results are very close to each other as well. When the spectrum is divided into 4 bands, WBCK-3 offers the best accuracy among the three WBCK models investigated.

Conclusions

In this work, a nongray-wall emissivity model is proposed within the framework of the Wide-Band Correlated K-distribution method. In addition, an absorption-coefficient-based strategy is developed to optimize the band intervals. Three 1D isothermal and homogeneous cases bounded by the fly-ash deposit, GH536, and soot deposit and a 3D fuel-air flame bounded by a rectangular enclosure bounded by fly-ash deposit coated walls are considered as test cases to evaluate the accuracy of WBCK-1, WBCK-2 and WBCK-3 for different numbers of bands up to 4 in the prediction of radiative heat transfer. The LBL and NBCK results are obtained and used as the benchmarks in the one-and three-dimensional cases, respectively. The following conclusions can be drawn from this work:

(1) WBCK-2 improves the accuracy of WBCK-1 when the number of bands is larger than 1. For 1-band WBCK (i.e., FSCK), WBCK-2 is more accurate than WBCK-1 in the pure CO2 case, while they show the opposite trends in the pure H2O and CO2-H2O case.

(2) When the entire spectrum is divided into 2 bands, WBCK-3 demonstrates the worst accuracy among the models and thermal conditions investigated.

(3) When the spectrum is divided into 4 bands, WBCK-3 is more accurate than WBCK-1 and WBCK-2 for low-temperature wall. For high-temperature wall, both WBCK-2 and WBCK-3 agree well with the benchmark.

(4) For the three types of typical wall material of nongray emissivity encountered in practical combustion systems, it is sufficient to divide the entire spectrum into 3 or 4 bands when using WBCK-3 for accurate predictions of radiative heat transfer.

  AbbreviationsFSCK Full-Spectrum Correlated K-distribution LBL Line-By-Line MSMGFSK Multi-Scale Multi-Group FSK NBCK Narrow-Band Correlated K-distribution RTE radiative heat equation SLW Spectral-Line-based Weighted-sum-of-gray-gases SNBCK Statistical Narrow-Band Correlated-K

∫

  𝜀 𝜂 𝐼 𝑏𝜂 (𝑇 𝑤 )𝛿(𝑘 𝑚 -𝜅 𝜂 (𝜙 0 )) 𝑓 𝑇 0 ,𝜙 0 ,𝑚 (𝑘 𝑚 ) 𝜂 𝑢,𝑚 𝜂 𝑙,𝑚 𝑑𝜂 = ∫ [ 𝜀 𝜂 𝐼 𝑏𝜂 (𝑇 𝑤 )𝛿(𝑘 𝑚 -𝜅 𝜂 (𝜙 0 )) 𝑓 𝑇 𝑤 ,𝜙 0 ,𝑚 (𝑘 𝑚 ) • 𝑓 𝑇 𝑤 ,𝜙 0 ,𝑚 (𝑘 𝑚 ) 𝑓 𝑇 0 ,𝜙 0 ,𝑚 (𝑘 𝑚 ) ] 𝜂 𝑢,𝑚 𝜂 𝑙,𝑚 𝑑𝜂 = 𝜀 𝑚 * (𝑔 0 )𝑎 𝑚,𝑤 (𝑔 0 )𝐼 𝑏,𝑚 (𝑇 𝑤 )

1

 1 Schematic diagram of band interval determination method. (a) Emissivity-based band interval (b)Absorption-coefficient-based band interval

Fig. 2

 2 Fig. 2 Schematic diagram to illustrate the absorption-coefficient-based band interval.

3 Fig. 4

 34 Fig. 4 Variation of TE predicted by different WBCK models and for different wall surface materials with the wall temperature for Case 1.

3 Fig. 5

 35 Fig. 5 Variation of EF predicted by different WBCK models and for different wall surface materials with the wall temperature for Case 1.

  Figure6shows the distributions of radiative source term for wall temperature of 500 K and boundary condition of BC-3, corresponding to conditions marked by the vertical cyan dashed lines in Figures4(c) and 5(c). Apart from the conclusions mentioned above, it can be seen that the 4-band WBCK-1 and WBCK-2 models show different trends of LE(x). To be more specific, the LE(x) values of WBCK-1 and WBCK-2 at the mid-point of computational domain are almost the same, while LE(x) of WBCK-1 and WBCK-2 near the wall are obviously different. This is the reason why the TE values of 4-band WBCK-1 and WBCK-2 are almost identical, while the EF value of 4-band WBCK-1 is slightly smaller in magnitude than that of 4band WBCK-2.

Fig. 6

 6 Fig. 6 Distributions of the radiative source term of Case 1 for 500 K wall temperature with BC-3 as the boundary material.

Fig. 7

 7 Fig. 7 The prediction of 2nd-band emissivity of two-band WBCK-1 and WBCK-2. The left frame displays the 2nd-band spectral absorption coefficient, the spectral Planck function and the spectral emissivity of BC-3. The right frame shows the reordered absorption coefficient and the 2nd-band emissivity of WBCK-1 and WBCK-2.

Fig. 8

 8 Fig. 8 The prediction of 4th-band emissivity of 4-band WBCK-1 and WBCK-2. The left frame displays the 4th-band spectral absorption coefficient and the spectral emissivity of BC-3. The right frame shows the reordered absorption coefficient and the 4th-band emissivity of WBCK-1 and WBCK-2.

Fig. 9

 9 Fig. 9 The prediction of 3rd-band emissivity of 4-band WBCK-3. The left frame displays the 3rd-band spectral absorption coefficient and the spectral emissivity of BC-3. The right frame illustrates the reordered absorption coefficient and the 3rdband emissivity of WBCK-3.3.2 The pure H2O caseThe second test case (Case 2) contains pure H2O as the radiating gas, whose thermal conditions are given Table1. The

The 2 -

 2 and 4-band WBCK-1 models display different trends of TE and EF for different boundary materials, as in the pure CO2 case. For BC-1, both TE and EF of 2-and 4-band WBCK-1 first decrease sharply and then remain close to zero as the wall temperature further increases. For BC-2, as shown in the middle and bottom rows of Figures 10(b) and 11(b), TE and EF of 2-and 4-band WBCK-1 first decrease in magnitude then increase in magnitude with the wall temperature increases. As for BC-3, both TE and EF of 2-and 4-band WBCK-1 first decrease and then oscillate with increasing the wall temperature. In contrast, TE and EF of 2-and 4-band WBCK-2 and WBCK-3 decrease nearly monotonically in magnitude as the wall temperature increases regardless of the wall material, which are similar to Case 1.

3 Fig. 10 3 Fig. 11

 310311 Fig. 10 Variation of TE predicted by different WBCK models and for different wall surface materials with the wall temperature for Case 2.

3 Fig. 12 3 Fig. 13

 312313 Fig. 11 Variation of EF predicted by different WBCK models and for different wall surface materials with the wall temperature for Case 2.3.3 The CO2-H2O mixture caseIn the final 1D test case (Case 3), the medium is a CO2-H2O mixture and the thermal conditions are also given in Table1. The gaseous medium consists of 10% CO2 and 20% H2O (balanced by N2) at 2500 K and 1 atm, while the wall temperature is varied from 300 K to 2000 K. Figures12 and 13display the variations of TE and EF with the wall temperature. The trends of TE and EF of 1-, 2-and 4-band WBCK-1, WBCK-2 and WBCK-3 are consistent with those observed in Case 2. The performances of 1-, 2-and 4-band WBCK-1, WBCK-2 and WBCK-3 are also similar to those in the pure H2O case (Case 2) as well. In other words, 1-band WBCK-1 is more accurate than 1-band WBCK-2 (or WBCK-3), the 2-and 4-band WBCK-2 models demonstrate better accuracy than the 2-and 4-band WBCK-1 models, and the 2-band WBCK-3 model is less accurate than the 2-band WBCK-2 model; however, WBCK-3 can become very accurate when the entire spectrum is divided into four bands, especially at low wall temperatures.

3 Fig. 14

 314 Fig. 14 Variation of TE with the number of bands for Case 3, in which the wall temperature is 300 K.

Fig. 15

 15 Fig. 15 Variation of TE with the number of bands for Case 3, in which the wall temperature is 1500 K.

16

 16

  (b).

Fig. 16

 16 Fig.[START_REF] Solovjov | Efficient cumulative wavenumber model of radiative transfer in gaseous media bounded by non-gray walls[END_REF] The radiative source term distributions along the enclosure centerline (0, 0, z) of the fuel-air flame predicted by NBCK (bounded by the black walls and BC-1), 1-band WBCK-1, 1-band WBCK-2, and 4-band WBCK-1, WBCK-2 and WBCK-3.

Figure 17

 17 Figure17displays the net heat fluxes along the centerline of a side wall, (1 m, 1 m, z), by NBCK (bounded by the black walls and BC-1), 1-band WBCK-1, 1-band WBCK-2, and 4-band WBCK-1, WBCK-2 and WBCK-3. There are obvious discrepancies between the NBCK results for the two boundary conditions (black wall vs. BC-1). For the black walls, the maximum flux along the line of (1 m, 1 m, z) is 20.5 kw/m 2 , which is consistent with the result of Porter et al.[START_REF] Porter | Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments[END_REF], while for BC-1, the maximum net flux is lower at 16.1 kw/m 2 . Similar to the radiative source term along the center line (0, 0, z), 1-band WBCK-1 and WBCK-2 predicted undistinguishable results but with large errors. When using 4 bands, WBCK-1 and WBCK-2 yield significantly better accuracy than their 1-band counterparts and their results are very close to each other as well. When the spectrum is divided into 4 bands, WBCK-3 offers the best accuracy among the three WBCK models investigated.

Fig. 17

 17 Fig. 17 The radiative heat flux along the line of (1 m, 1 m, z) predicted by NBCK (bounded by the black walls and BC-1), 1-band WBCK-1, 1-band WBCK-2, and 4-band WBCK-1, WBCK-2 and WBCK-3.

  𝜂 𝑢,𝑚 and 𝜂 𝑙,𝑚 are the upper and lower limits of mth band, 𝐼 𝑏,𝑚 (T 𝑃 ) (=∫ 𝐼 𝑏𝜂 (T 𝑃 )d𝜂 𝑔 𝑚 , which represents the fraction of spectrum whose spectral absorption coefficient is lower than an absorption coefficient variable 𝑘 𝑚 , is expressed as:

	𝜂 𝑢,𝑚 𝜂 𝑙,𝑚	) is the spectrally-integrated
	Planck function of the mth band.
	The cumulative wide-band k-distribution function of the mth band, 𝑔 𝑇 𝑃 ,𝜙,𝑚 (𝑘 𝑚 ) = ∫ 𝑓 𝑇 𝑃 ,𝜙,𝑚 (𝑘 𝑚 ′ 𝑘 𝑚 ′ )𝑑𝑘 𝑚 0

Acknowledgments

The authors are grateful for the former support of "National Natural Science Foundation of China "(Grant No.50606004 ) on the radiative heat transfer modelling topic, and the support of "Overseas Expertise Introduction Project for Discipline Innovation "(111 project, Grant No. B08009 ) for international academic exchanges.