，，，*，

1.College of Energy and Power Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China；

2.Aviation Key Laboratory of Science and Technology on Aero Electromechanical System Integration，Nanjing 211106，P.R.China

Abstract: Soot，a product of insufficient combustion，is usually in the form of aggregate.The multi-scattering of soot fractal aggregates has been proved to play an important role in studying the soot radiative properties，which is rarely considered in predicting the radiative heat transfer in combustion flame.In the present study，based on the weighted sum of gray soot fractal aggregate（WSGSA）model，which is used to predict the temperature field and soot aggregates in turbulent diffusion flame，the flame temperature distribution and soot volume fraction distribution under the conditions of the model without considering radiation，the default radiation model in Fluent software and the WSGSA model are calculated respectively.The results show that the flame temperature will be seriously overestimated without considering radiation and the maximum relative discrepancy of flame centerline temperature is about 64.5%.The accuracy will be improved by the default radiation model in the Fluent software，but the flame temperature is still overestimated and the maximum relative discrepancy of flame centerline temperature is about 42.1%.However，more satisfactory results can be obtained by the WSGSA model，and the maximum relative discrepancy of flame centerline temperature is no more than 15.3%.Similar conclusions can also be obtained in studying the temperature distribution along different flame heights.Moreover，the soot volume fraction can be predicted more accurately with the application of the WSGSA model.Both without considering radiation and using the default radiation model in the Fluent software will result in the underestimating of soot volume fraction.All the results reveal that the WSGSA model can be used to predict the temperature and soot aggregates in the C2H4/air turbulent diffusion flame.

Key words：radiative heat transfer；WSGSA model；soot radiation；turbulent diffusion flame；soot aggregate

0 Introduction

Radiative heat transfer is critical to the research on combustion and heat transfer mechanisms of the combustion flame［1］.As the main product of insuffi?cient combustion of hydrocarbon fuels，the soot can emit radiant energy strongly and continuously in the whole infrared spectrum［2］，which has been proved to be the second largest factor of greenhouse effect and global warming，second only to carbon diox?ide［3］.Experimental research proves that the ability of emitting radiant energy of soot is far stronger than that of participated gas at the same temperature［2］.Although the volume fraction of soot is small，the ability of emitting radiant energy cannot be ignored and the spectral radiant intensity of flame is mainly determined by the soot.Therefore，establishing an efficient and accurate calculation model to study the radiative properties of the soot and gas is of great significance for experimental design and pollutant emission control.

The radiative properties of high temperature gases strongly depend on parameters such as wave number，temperature and pressure，which bring great difficulties to solve the radiative transmission equation.Generally，the common numerical calcula?tion models used to predict the radiative transfer in the gas medium include line-by-line（LBL）model，band model（BM）and global model（GM）.The LBL model shows the highest calculation accuracy and the spectral variable information can be fully taken into account by integrating over all wave num?ber intervals.However，this model usually needs a huge spectral database，which has the slowest cal?culation speed and high requirements for calculation resources，therefore，it cannot be applied to engi?neering calculation.The BM model considers both calculation efficiency and calculation accuracy.The main idea of BM model is replacing its spectral radi?ative characteristic parameters by the average value of radiative characteristic parameters in spectral inte?gration interval.The GM model is an effective bal?ance between calculation efficiency and calculation accuracy.In the actual process of radiative heat transfer calculations，we tend to care more about the total radiative heat flux and radiative source within the whole wave number.The GM model can integrate the radiative characteristic parameters in the whole spectral range and solve the radiative transfer equation only for very few times.More?over，it can be combined with different methods for solving the radiative transmission equation and can be effectively coupled with computational fluid dy?namics software.Generally，the GM model mainly includes Planck average absorption coefficient mod?el，weighted-sum-of-gray-gases（WSGG）model，spectral-line-based WSGG（SLW-WSGG）model，full-spectrumk-distribution（FSK）model and vari?ous extended models based on these models.The WSGG model，first proposed in Ref.［4］in 1967，is widely used for its high efficiency，simplified cal?culation process and relatively high calculation accu?racy.The model can be combined with any form of radiative heat transfer equation and it can realize the coupling solution of radiation，convection and heat conduction by embedding user-defined functions in?to computational fluid dynamics software，which makes the WSGG model develop rapidly.Through the continuous efforts of many scholars，from the initial model parameters only applicable to specific H2O/CO2partial pressure ratio［5］，a variety of WS?GG model parameters applicable to different H2O/CO2partial pressure ratios［6］，oxygen-rich combus?tion［7-8］and high-pressure combustion［9］have been gradually developed.With the release of the latest molecular spectrum database HITEMP2010，many new WSGG model parameters obtained by fitting emissivity based on HITEMP2010 database have fully expanded the applicable temperature range and pressure range［10-11］.

The soot produced by hydrocarbon fuel com?bustion usually exists in the form of fractal aggre?gate and the aggregate may contain hundreds of pri?mary particles.The primary particle size range is generally between 10 nm and 60 nm，and may reach 300 nm in extreme cases.To accurately calculate the radiative properties of fractal aggregate of soot，many numerical calculation models have been developed，such as generalized multi-particle MIE theory（GMM）model［12］，multi-sphere?T-matrix（MSTM）［13］and method of moment（MOM），etc.The approx?imate models include Rayleigh-Deybe-Gans-fractalaggregate（RDG-FA）model，discrete Diople ap?proximation（DDA），finite difference time domain（FDTD），and the equivalent-sphere model.

Recently，considerable work has reported the studies of the radiative heat transfer in the gas and soot mixture［7-8］.The soot is usually assumed to be single and dispersed，and the effect of soot aggrega?tion is rarely considered.In fact，the soot aggrega?tion has been proved to affect the radiative proper?ties of soots obviously.In Refs.［1，14］，the influ?ences of soot aggregation on the radiative heat trans?fer in homogeneous gas-soot mixtures were also studied on the basis of the FSK model and GMM theory，and the results revealed that ignoring the ef?fect of soot aggregation would overpredict the radia?tive heat transfer properties of the mixture.There?fore，to predict the radiative heat transfer in the gassoot mixture including aggregates more accurately，the effect of soot fractal aggregation should be con?sidered.Therefore，a model，named weighted sum of gray soot fractal aggregate（WSGSA）model，is developed by combining the features of the FSK model with WSGG model.More details can be achieved in Refs.［1，14］.By using the WSGSA model the heat transfer properties of the C2H4/air turbulent diffusion flame are studied in the present work.The flame temperature distribution and soot volume fraction distribution are calculated respec?tively under the conditions of the model without con?sidering radiation，the default radiation model in Fluent software and the WSGSA model.Finally，the accuracy of the WSGSA model is verified by combined with the WSGG model in the Fluent soft?ware.

1 Physical Model

In this paper，the coaxial C2H4/air turbulent dif?fusion flame experiment carried out by Kent et al.［15］is taken as the research object.As a classic experi?ment measuring the axial temperature distribution，radial temperature distribution and soot volume fraction distribution，it has been used as the bench?mark to verify the numerical calculation results for many times in the past decades［16-17］.The calcula?tion domain model is shown in Fig.1.The fuel pre?heated to 322 K（pure C2H4）is shot into the air at an average speed of 52 m/s through a nozzle with an inner diameter of 3.0 mm to form a diffusion flame.The Reynolds number at the nozzle outlet isRe=14 660.Since the flame has two-dimensional axisymmetric characteristics，half of the calculation domain is taken for non-uniform mesh division to improve the calculation efficiency.Moreover，the mesh encryption is carried out for the fuel nozzle outlet and the area near the axis of symmetry.The mesh division results are shown in Fig.2，whererrepresents the radius andZthe direction of the flow.

Fig.1 Physical model and geometric structure diagram of research object[14]

Fig.2 Non-uniform grid division diagram in computational domain

2 Computational Model

2.1 Governing equation

The standardk-εturbulence model is adopted.The transport equations of mass conservation，mo?mentum conservation，energy conservation，turbu?lent kinetic energy and turbulent kinetic energy dissi?pation rate can all be given in the general form of Eq.（1）.

whereρ，t，?，Γ?andS?denote the density，the time，the scalar，the diffusion coefficient and the source term，respectively，which are summarized in Table 1，whereμrepresents the velocity of the com?ponent，Prthe Planck number，εthe emissivity，σthe spectral scattering coefficient，andfthe fraction of the mixture.

2.2 Combustion model

The combustion model in this paper is a steady laminar flame surface model，and the turbu?lent flame can be regarded as a set of one-dimen?sional laminar structures.To express the flame bythe isosurface of mixed components，the left and right components are assumed to behave the same diffusion coefficient.The coefficient values ofσt，CgandCdare set as 0.85，2.96 and 2.0，respec?tively.The premixed combustion is modeled byCequation model and the reaction process is repre?sented by scalarC.The transport equation of sca?larCis shown in Table 1.The value range of sca?larCis set as［0，1］，where 0 represents un?burned reactant and 1 represents burnt reactant.The effects of turbulent pulsation on flame temper?ature，composition and density are established by using probability density function（PDF）.The GRI3.0 detailed chemical reaction mechanism［18］is adopted.

Table 1 Corresponding scalar,diffusion coefficient and source term in governing equations

2.3 Soot generation model

The Moss-Brookes two-equation model in the semi-empirical model is used to predict the soot generation.The model obtains the soot volume fraction distribution by solving the transport equa?tion of soot nucleation aggregation and soot mass fraction［19］

whereYsootrepresents the soot mass fraction，Mthe soot nucleation aggregation，Nnorm=1015，the normalized atoms nucleation concentration（=N/ρNnorm），μt/σsootthe diffusion coefficient of soot mass fraction，andμt/σnucthe diffusion coefficient of atom nucleation concentration.

It is assumed that C2H2and C2H4are the chemi?cal components of soot nucleation，surface growth and other chemical reactions.The oxidation model is Lee model，in which hydroxyl［OH］is used as the oxidant and the concentration of locally balanced hydroxyl［OH］and equilibrium oxygen atom［O］can be respectively expressed by

2.4 Radiation model

2.4.1 Radiative transfer equation

For a one-dimensional dispersive mixed medi?um with absorption，emission and scattering proper?ties composed of participating gases（such as H2O，CO2，etc.）and soot，the internal spectral radiative transfer process can be described by the one-dimen?sional spectral radiative transfer equation［19］

whereIλ(z，s) is the spectral radiation intensity in the direction at positionz，Ib，λ(Tm，z) the blackbody spectral radiation intensity at positionzand the tem?peratureTm，Φ(s，s)the scattering phase function of the mixed medium，Ω'the solid angle，andκm，λthe spectral absorption coefficient in the mixed medium.κm，λ=κgas，λ+κsoot，λ，κgas，λandκsoot，λrepresent the spectral absorption coefficients of the participating gas and soot respectively.σm，λrepresents the spec?tral scattering coefficient of the mixed medium.The boundary condition of non-grey boundary is［20］

whereεwis the wall emissivity，εw=1 the black?body wall，Ib，λthe blackbody spectral radiation inten?sity at the wave numberλ，Twthe temperature of the wall，andnthe outer normal direction vector of the wall.zwandsare the location coordinates.

According to the WSGG model theory，soot can also be regarded as non-gray medium，just like participating gases.Therefore，the radiative transfer equation and boundary conditions in the mixed medi?um containing participating gas and soot can be writ?ten as follows［21］

whereNrepresents the number of grey medium se?lected，εthe emitting rate of the boundary，andκm，ithe absorption coefficient of theith gray mixture consisting of gas and soot fractal aggregates.κm，i=κgas，i+κsoot，i，κgas，iandκsoot，irepresent the spectral absorption coefficients of theith participating gas and soot，respectively.σm，irepresents the spectral scattering coefficient of theith gray mixture consist?ing of gas and soot fractal aggregates.am，i(Tm) rep?resents the weight factor of theith gray mixture con?sisting of gas and soot fractal aggregates at the Gaussian integral point，am，i(Tm)=agas，i(Tm)×asoot，i(Tm).agas，i(Tm) represents the weighting factor of theith gray gas，andasoot，i(Tm)the weighting fac?tor of theith soot.In this paper，the discrete ordi?nates model of Fluent software platform is used to solve the radiative transfer equation，in which the gas radiation model and the soot radiation model are described below.

2.4.2 Gas emission model—WSGG

The common participating gases in combustion products include H2O and CO2.According to the WSGG model theory，the absorption coefficient of theith gray gas and the corresponding weight factor can be obtained by fitting the calculation results of the LBL model.Therefore，the emissivityεgasin the pure participatory gas medium can be expressed as［19］

whereNgasrepresents the number of non-gray gases selected，(T) the weight factor related to tem?perature，Pgasthe partial pressure of gas，Tthe gas temperature，andLthe thickness of the medium.

2.4.3 Soot aggregates radiationmodel—WSGSA

Based on the WSGG model and the basic idea of full spectralkdistribution，the spectral radiative properties of soot aggregates，e.g.absorption cross section and scattering cross section，are predicted by the RDG-FA model for different arrangements at first.Then，these spectral radiative properties are reordered in smoothly-varying g-space according the principle of FSK model，and the values at the Gaussian-Legendre quadrature integral points in the g-space are obtained.Finally，according to principle of the WSGG model，the relationship between the radiative properties at the Gaussian-Legendre quadrature integral points and the geometric charac?teristic parameters of the soot aggregates，e.g.pri?mary particle number，are obtained by data fitting.

The cluster-cluster aggregation（CCA）mod?el，developed by Mackowski et al.［13］and widely used in studying the aggregation of soot and aero?sols，is proposed to study the soot aggregation pro?cess in the present work.The details of the CCA model is available in Refs.［1，14］.According to the fractal theory，the morphology and construction of the aggregate can be described as

whereNsis the total number of the primary mono?mers in the aggregate，dpthe mean diameter of the monomers，kfthe fractal prefactor，Dfthe fractal di?mension，Rgthe root mean square radius that quanti?fies the overall size of the aggregate and is called the gyration radius，andrithe distance from theith sphere to the center of the aggregate mass.The rela?tionship between the gyration radiusRgand the num?ber of monomersNsin the aggregates can thus be connected by the fractal dimensionDfand fractal prefactorkf.

According to Refs.［1，14］，the values for the fractal dimensionDfof the soot fractal aggregates al?most fall within the range of 1.6—1.9，the particle size of the soot is set as 33 nm to simplify the calcu?lation process and make the problem mathematically trackable，and the total number of the primary mono?mers in the soot fractal aggregates is set in the range of 20—300.

whereNsootrepresents the number of gray soot ag?gregates selected，Tthe gas temperature，Lthe thickness of the medium，andthe weight factor related to temperature，which can be ex?pressed as

wherefv，irepresents the aggregate volume concen?tration of soot，andNs，ithe number of particles in the aggregate of soot.In addition，the scattering phase function Henyey-Greenstein（H-G）of soot ag?gregates can be expressed as

According to the previous research results of our team，the radiative properties of soot aggregates are correlated with the number of soot monomers in the aggregates［22-25］，and the absorption coefficient is linearly correlated with the number of particles in the aggregate.Therefore，the absorption coefficient of soot aggregatescan be expressed as

wherepjandp0，jrepresent the polynomial coeffi?cients of thejth soot aggregate.Different from the absorption coefficient，the aggregate scattering coef?ficientasymmetry factorand the number of monomer particles in soot meet multiple power correlation are

wherep0，j，qj，iandoj，ican be obtained through data fitting.For detailed research process，please refer to Ref.［1］.

2.5 Default model in Fluent software

In this paper，the high-temperature gas absorp?tion coefficient model and soot absorption coeffi?cient model in the Fluent software are used to verify the WSGSA model，which is recorded as the de?fault model.The high-temperature gas radiation model in the Fluent software is based on the WSGG model developed by Smith in 1982［14］，and the total absorption coefficient is calculated by calculating the total emissivity

whereS=3.6V/Ais the optical thickness，Vthe to?tal volume of the calculation domain，andAthe to?tal surface area of the calculation domain.

The default soot absorption coefficient model in the Fluent software is

whereb1=1 231.4 m2/kg，bT=4.8e-4K-1.ρsootandYsootare the density and mass fraction of soot aggre?gate.

So，the absorption coefficient of the gas-soot mixture can be expressed as

whereκgandκgare the absorption coefficients of the gas and the soot，respectively.

3 Simulation Results and Discus?sion

The flame temperature distribution and soot volume fraction distribution under the condition of the model without considering radiation，the default radiation model in Fluent software and the WSGSA model are calculated respectively in the coaxial C2H4/air turbulent diffusion flame.

3.1 Grid independence validation

To verify that the calculated results are inde?pendent of the number of grids，this section carries out numerical calculations with three sets of grids with the number of 12 736，31 096 and 78 842，re?spectively.The obtained axial velocity and tempera?ture distributions along the axis of symmetry are shown in Figs.3（a）and（b），respectively.It is clear that there is a slight difference in temperature be?tween the 12 736 grids and the other two sets of grids and there is almost no difference between the results of 31 096 grids and 78 842 grids.To increase the computational efficiency，31 096 grids are used for subsequent research.

Fig.3 Grid independence verification

3.2 Results analysis

Fig.4 shows the flame centerline temperature compared with the experimental values.It can be seen that the prediction values of the WSGSA mod?el developed in this paper are in good agreement with the experimental values and the maximum rela?tive discrepancy is no more than 15.3%.When the axial distance is within 0.17 m，all the three models have a good agreement with the experimental val?ues.However，when the axial distance exceeds 0.17 m，the flame temperature will be seriously overestimated without considering radiation，the maximum temperature will reach 2 191 K，which is 585 K away from the experimental values and the maximum relative discrepancy reaches 64.5%.The default model in the Fluent software and WSGSA model has almost the same prediction values when the axial distance is within 0.33 m，but the predic?tion values will also be overestimated by the default model when the axial distance exceeds 0.33 m，and the maximum relative discrepancy against the exper?imental values is about 42.1%，which is better than that without considering radiation.

Fig.4 Axial temperature distribution of flame center line

Figs.5（a，b，c）are the temperature distribution cloud without considering radiation，calculated by the default model in the Fluent software，and ob?tained by the WSGSA model，respectively.

Fig.5 Temperature distribution cloud under different radia?tion model conditions

The solid black line is an isothermal line with a temperature equal to 1 800 K and the temperature surrounded by the solid black line is more than 1 800 K.It can be seen that when radiation is not considered，the high temperature range is the larg?est，followed by the Fluent software default model and the high temperature range is the smallest by WSGSA model.So，it is obvious that when radia?tion is not considered，not only the flame tempera?ture will be overestimated，but the high temperature range of the flame will also be overestimated.

Similar conclusions can be obtained from the ra?dial flame temperature distribution at different flame height positions.Fig.6 shows the radial flame tem?perature distribution at flame heightZ=0.138，0.241 5，0.345 m，respectively.It can be seen from Fig.6 that the prediction radial flame temperature distribution by WSGSA model has the best agree?ment with the experiment values at different flame height positions.When radiation is not considered，the radial flame temperature will be overestimated at all heights，especially at flame heightZ=0.345 m.The predicton values of the default model are better than the model which does not consider ra?diation but are worse than WSGSA model.

Fig.6 Radial temperature distribution at different flame heights

Fig.7 shows the radial soot volume fraction dis?tribution at the flame heightsZ=0.138，0.241 5，0.345 m，respectively.The soot volume fraction predicted by WSGSA model is in good agreement with the experimental value.Especially atZ=0.345 m，the maximum relative error is no more than 30%，which is more accurate than the results obtained without considering the radiative heat trans?fer or using the radiation with Fluent default model.The soot volume fraction will be underestimated in the other cases，especially without considering the radiation.The likely explanation may be that radia?tive heat transfer properties of soot can be accurately described by the WSGSA model，so more satisfac?tory flame temperature distribution can be obtained.As far as we know，the soot volume fraction is closely related to the flame temperature and the high temperature will enhance soot oxidation.Therefore，the soot volume fraction predicted by the WSGSA model is more satisfactory for more accurate temper?ature obtained.

Fig.7 Radial soot concentration distribution at different flame heights

4 Conclusions

The WSGSA model is used to predict the tem?perature distribution and soot volume fraction in the coaxial C2H4/air turbulent diffusion flame.The re?sults show that WSGSA model can not only be cou?pled with WSGG model and embedded in Fluent software platform，but also can more accurately pre?dict the temperature distribution and soot concentra?tion distribution of the flame，compared with cases without considering radiative heat transfer or with default model in Fluent software.The conclusions drawn from the results are as follows：

（1）The results reveal that the flame tempera?ture will be seriously overestimated without consid?ering radiation and the maximum relative discrepan?cy of flame centerline temperature is about 64.5%.The accuracy will be improved by the default radia?tion model in the Fluent software，but the flame temperature is still overestimated and the maximum relative discrepancy of flame centerline temperature is about 42.1%.However，more satisfactory results can be obtained by the WSGSA model，and the maximum relative discrepancy of flame centerline temperature is no more than 15.3%.

（2）The high temperature region will be re?duced obviously with using the WSGSA model con?sidered，which is conducive to decline the thermal protection requirements in an actual combustion de?vice.

（3）The soot concentration distribution will be predicted more accurately with considering the radia?tive heat transfer，especially the WSGSA model be?ing used.

The WSGSA model shows a good perfor?mance in predicting the temperature field and soot volume fraction of flame，which can be further ap?plied to the prediction of the temperature and soot in the carbon and hydrogen flame soot.