Two-flux method for radiation heat transfer in anisotropic gas-particles media

Science in China Ser. E Engineering & Materials Science 2004 VoL.47 No.6 625- 640625Two-flux method for radiation heat transfer inanisotropic gas-particles mediaWANG Fei', CEN Kefal', T. Girasole2, A. Garo2, G. Grehan2 & YAN Jianhua'1. Institute for Thermal Power Engineering, Zhejiang University, Hangzhou 310027, China;2. LESP, UMR 6614/CORIA, CNRS, Universite et INSA de Rouen, BP12, 76801, Saint Etienne du Rou-vray, FranceCorrespondence should be addressed to Wang Fei (email: wangfei@cmee.zju.edu.cn)Received June 8, 2004Abstract Two-flux method can be used, as a simplification for the radiative heattransfer, to predict heat flux in a slab consisting of gas and particles. In the originaltwo-flux method (Schuster, 1905 and Schwarzschild, 1906), the radiation field wasassumed to be isotropic. But for gas-particles mixture in combustion environments, thescatterings of particles are usually anisotropic, and the original two-flux method givescritical errors when ignoring this anisotropy. In the present paper, a multilayer four-fluxmodel developed by Roze et al. (2001) is extended to calculate the radiation heat flux in aslab containing participating particles and gas mixture. The analytic resolution of theradiative transfer equation in the framework of a two-flux approach is presented. Theaverage crossing parameter ε and the forward scattering ratio ζ are defined to describethe anisotropy of the radiative field. To validate the model, the radiation transfer in a slabhas been computed. Comparisons with the exact analytical result of Modest (1993) andthe original two-flux model show the exactness and the improvement. The emissivity of aslab containing flyash/CO2/H2O mixture is obtained using the new model. The result isidentical with that of Goodwin (1989).Keywords: radiative transfer, two-flux, anisotropic.DOI; 10.1360/03ye03141 IntroductionTwo-flux method is considered as a simple model and can be used to predict heatflux in 1-D radiative heat transfer. In the original two-flux methodu2, the radiation fieldwas assumed to be isotropic. But for gas-particles mixture in combustion environments,the scatterings of particles are usually anisotropic, and the original two-flux methodgives critical errors when ignoring this anisotropy. It is desirable to have a two-flux ra-diative transfer model to be applied regardless of the degree of anisotropy and withoutassuming the symmetry condition.中国煤化工A multilayer four-flux method has been deve.MYHCNMHGandcolli-Copyright by Science in China Press 2004626Science in China Ser. E Engineering & Materials Science 2004 Vol.47 No.6 625- -640mated fluxes in a slab containing scattering, emitting and absorbing particlesl- 5. Anadvantage of this approach is that the anisotropy of the particles scattering can be intro-duced. In the present paper, the four-flux model is extended to calculate the radiationheat flux in a slab containing participating particles and gas mixture. As the collimatedbeam in the four-flux model is not necessarily present in radiation heat transfer, themultilayer four-flux model is degenerated in a two-flux method.The analytic resolution of the radiative transfer equation in the framework of thetwo-flux approach is presented. The average crossing parameter ε and the forward scat-tering ratio ζ are defined to describe the isotropy of the radiative field. The optical prop-erties of the particles are determined using Lorenz- Mie theory. To validate the model,the radiation transfer in a slab has been computed. Comparisons with the analytical re-sult of Modestbl and the original two-flux model show the exactness and the improve-ment. The emissivity of a slab containing flyash/CO2/H2O mixture is obtained using thenew model, which is identical with that of Goodwin72 Improvement for original two-flux modelAs shown in fig. 1, for an absorption, scattering and emission participating medium,radiation transfer equation (RTE) in a specified direction θ (referred to as μ = cos(θ ) inthis paper), is as followsl8!: (Here, the subscripts of wavelength are omitted because allthe parameters are in a given wavelength)μddI(z,μ)=-(k, +k。+s)[(z,μ)+(k, +k)1,(z)+号[ "I(z,μ')p(H,μl)du'， (1)lzwhere kp is the absorption coefficient of the particle, kg is the absorption coefficient ofthe gas, and s is the scattering coefficient of the particle.(Z)(O)J2一→H2O, CO2●62 T2particles, TE. Ti中国煤化工Fig. 1. Geometry of a sinYHCNMHGCopyright by Science in China Press 2004Two-flux method for radiation heat transfer in anisotropic gas- particles media627This equation has the boundary condition: the slab is subject to two incomingknown isotropic fluxes(Jσ atz=0and Jz atz=Z).Schuster'" and Schwarzschild2] have developed the original two flux method. Intheir two-flux method, the radiation field was divided into the forward (0≤μ≤1) andthe backward (-1≤μ≤0), and for each hemisphere, the mean intensities I*(z) andI (z) were defined, as shown in fig. 2.(Z)(O)1(z,0)Jz一>I (z)52 T2&1 T<0Fig. 2. Definition of the mean intensity.Mathematically, I +(z) and I ~(z) are the arithmetic means of I (z, μ) over therespective forward > 0) and backward (μ < 0) hemispheres. The“mean intensities"are defined as .r*(z)=. [(,u)du,I(2)=. [_(,u)du,In the original two flux model9), an approximation has been made:(4)f"(au)udu 1J] TJi5R右MoFig. 3. Energy balance in a sub-layer.Thenq (0)=T.Jz +R.J +M,(39)According to eq. (38), we findq (0)=_π(C +C203 +1,).(40)ε~ (0]Due to the boundary condition of the average crossing parameter,ε (0)=2,we haveq (0)=(C + Cz[Q3)JZ +(C12 +C22(3)J古+πC3 +πC293 +π1p.(41)q (0)=1 ax32|J +o.-exp(-. .Z)+ C3 exp(VA . z|片D2 D2D2+Lea,exp(-√A .Z)-1]+m/65. -a, exp(√A .Z)+a32]D2 L+nL.[exp(vA. Z)-a3°exp(-A .z)].(42)DComparing eq. (42) with eq. (39), we can seeT=-中国煤化工(43)MYHCNMHGCopyright by Science in China Press 2004Two-flux method for radiation heat transfer in anisotropic gas- particles media635(44)R= [cx(VA .Z)-exp(-A .z)].D2M=p( - a3)[exp(VA Z)+asexp(-JA .Z)-1-a],(45)where e, =πI，T is the hemispherical transmittance, and R is the hemispherical le-flectance. Defining the hemispherical emissivity of the slab E asE=M(46)e,eq. (45) changes toC.3[exp(VA. Z)+asexp(-JA .z)-1-as].(47)By simple algorithm, we can obtainE=A=1-T- R,where A is the absorptance of the slab, which equalsA=1-T-R.Because we have assumed that the boundary is a diffuse-grey surface and theincident radiation is diffuse completely, according to Kirchhoff" s lawts, the hemis-pherical emissivity should equal the hemispherical absorptance of the slab. So, amongeqs. (43), (44) and (47), only two of them are independent.3 Calculations for η and ζIn this model, the calculation for η and ζ is the key point. Roze and Girasolecalculated the transmittance and the reflectance of a slab with Monte Carlo method, byscanning over most possible values of the scattering albedo a, the asymmetry parameterg (due to Lorenz-Mie theory) and the optical thickness b. The average crossingparameter and the forward scattering ratio in the four-flux were taken as adjustable pa-rameters. By comparing the results of four-flux model with Monte Carlo simulation, theoptimal values for the average crossing parameter and the forward scattering ratio werededuced. Because the emittance of the slab E is dependent on the slab transmittance Tand the slab reflectance R, those results can be used in this paper directly.The details about these calculations can be found in ref. [10]. It should be men-tioned that the ε in ref. [10] is the equivalent average crossing parameter η defined inthis paper.中国煤化工MYHCNMHGwww.scichina.com636Science in China Ser. E Engineering & Materials Science 2004 Vol.47 No.6 625- -6404 Validation for the modelTo validate the accuracy and improvement of the two- flux model presented in thispaper, we consider the problem of a single layer of non-absorption gas with particles at aconstant temperature bounded by black walls at zero temperature. The optical index ofthe particles is m= 2 - i and their radius is r= 5 μm. The entire cloud has a density of10'0 particles/m' and the Mie calculations are carried out for the typical wavelength ofλ= 3.1416 μm. The results are compared to the original two-flux model and the exactsolutions calculated with series expansion method by Modesto.1.00.8Original two-fux (isotropic)Present model0.6Modest0.4 |r=0.1τ=10.0).2 t0.0-0.6ro=1.0- -0.8-1.00.20.4Fig. 4. Comparisons with other models.Fig. 4 shows the comparisons of the local radiative heat flux rates in the isothermalslab calculated by the present two-flux method, the original two-flux model and the ex-act one by Modest for optical thickness To= 0.1, 1.0, and 10.0. The result of the originaltwo-flux model has serious difference with the exact value, especially when the opticalthichness is great. That is resonable because the latter has considered the anisotropicy ofthe particle scattering. And the difference between the result of the present two-fluxmodel and the exact result is negligible. These review the improvement to the originalmodel and the accuracy of the model in this paper.5 The radiative transfer calculation of flyash and gas mediaThis two-flux model is also applied in combustion field. The test case, comparedwith the results of Goodwin'", is an isothermal slab with transparent boundaries, whichcontains water vapor (H2O), carbon dioxide (CO2), and flyash. The dimensions of theslab, the temperature, the partial pressures of the中国煤化工oading arechosen to approximate typical values for a pulveriMHCNMHGahigh-ashCopyright by Science in China Press 2004Two-flux method for radiation heat transfer in anisotropic gas- particles media637coal, which are based on values suggested by Wall et al.". The values of the parametersare given in table 1.Table 1 The parameters for the test caseTemperaturePwatPrnoPcorSlab thicknessFlyash mass burden1500 K1 atm0.08 atm0.145 atm3m.6.5X 10-3 kg/kgTable 2 shows the models used in calculations of Goodwin and this paper respec-tively.Table2 The models in two calculationsGoodwin?lPresent two -fluxMethod to solve the equation ofthe Case Normal Model tech- the two _flux model in the presentradiative transferniquell21paperAnisotropic scatteringwith a scaled scattering coefficient average crossing parameter ε andby Gupta et al.!"3]forward scattering ratio 5 [10]5.1 Optical constants of the flyashThe optical contants of flyash presented by Goodwin 14 are employed in the presentcalculations. The experimental data of flyash optical constant are approximated bypiecewise linear function of wavelength. The real part n is written asn=1.50.5<λ < 6.0 μm,n=1.5- 0.35(入- 6.0) .6.0<λ < 8.0 μm,n=0.8+ 0.5(2- 8.0)8.0<λ<11.0 μ m,n=2.3-0.5(N- 11.0)11.0< λ< 12.0 μm;the image part k is .k= 10-4.6+2.2(2-0.5)0.5<λ<1.0 μm,k=10-3.51.0<λ<4.0 μm,k= 10-3.5+(a-4.0)4.0< λ<5.0 μ m,k= 102.5+0.24(入-5.0)5.0<λ <7.5 μm,-1.9+1.8(0-7.5)7.5<λ <8.5 μm,k=10-0.185<λ<10.5 um.中国煤化工k = 0107330-10.5)YHCNMHGwww.scichina.com638Science in China Ser. E Engineering & Materials Science 2004 Vol.47 No.6 625- -6405.2 Size distribution of the flyashThe size distribution of flyash used by Goodwin was obtained from measurementsin the laboratory on the flyash collected at a middle-field location of an electrical powerplant electrostatic precipitator. It was assumed to be log-normal in form, so that thefraction of particles f (D) with diameter between D and D + dD wasf(D)=V(2x)Dσ=exp[-(InD-u)2 12σ2],u=-0.214， σu = 0.747.5.3 The absorption coefficient and the scattering coefficient of the flyash cloudThe absorption efficiency Qabs,D and the scattering efficiency Qsca,D for a singleparticle of diameter D can be computed by the Lorenz-Mie theory. The cloud absorptioncoefficient Kp and the cloud scattering coefficient s are the sums of the contribution fromeach diameter fraction' 15]. ThereforeK,=15B![°f(DQabs,dD,Jo~ Ds=1.5.B.[°f(D)Qxu pdD,Jo Dwhere B is the flyash mass burden, and ρ is the density of the flyash.5.4 The values of η and ζRef. [10] gave the polynomial formulations for the equivalent average crossingparameter η and the forward scattering ratio ζ. For diffuse ilumination, the formulationsare as follows:η(a,b,g)=2+ 2之Caij;(a' -1)b'g*,旨j=0k=0k≠lζ(a,b,g)=:亡之它, Zβja'b'g*.i=0j=0k=0The coefficients a and β are shown in tables 4 and 5 in ref. [10].5.5 The absorption coefficient of the gasThe spectral absorption coefficients of the water vapor and carbon dioxide gas werecalculated using a narrow-band model program' 0. The absorption coefficient for the gaswas computed as中国煤化工MYHCNMHGCopyright by Science in China Press 2004Two-flux method for radiation heat transfer in anisotropic gas- particles media639K。=-ln(r,)/L,where τg is the narrow-band transmittance of the gas mixture, and L is the pathlengthused to calculate τg. According to Goodwin'", L equals twice the slab thickness, which isthe geometric mean beam length in the optically-thin, non-scattering limit.H2O, CO, ash0.8-日0.6-0.4-H2O, CO2).2 ta)|0.0461021.0H2O, CO2, ash0.80.60.40.2H2O,CO2(b).012Wavelength/umFig. 5. The emissivity of the slab (with flyash and without flyash). (a) The results with model in the present paper;(b) the results of Goodwin'!.In fig. 5, the spectral emissivities of the slab (with flyash and without flyash) areshown. Fig. 5(a) is the results calculated with the present two-flux model. Fig. 5(b) is theresults of Goodwin'!. Because different gas models are introduced in the twocalculations, the results are not the same completely, but the similarity of the two resultsis enough to iluminate the applicability of the pr中国煤化工ombustionradiative transfer.YHCNMHGwww.scichina.com640Science in China Ser. E Engineering & Materials Science 2004 Vol.47 No.6 625- -6406 ConclusionIn this paper, the analytic resolution of a two-flux approach concerning the anisot-ropy is presented, which is developed based on the multilayer four-flux method by Rozeet a1.t0. The average crossing parameter and the forward scattering ratio are introducedto describe the scattering anisotropy. To validate the model, the radiation transfer in aslab has been computed. Compared with the analytical result of Modestb and the origi-nal two-flux model, the model shows the exactness and the improvement. The emissivityof a slab containing flyash/CO2/H2O mixture is obtained using the new model. The re-sult is identical with that of Goodwin!7The further research is to develop the two-flux model in this paper into muli-fluxmodel, in order to solve the three -dimensional radiative transfer problem.Acknowledgements The authors would like to thank the sponsorship by le Ministere de la Recherche de France,the Programme Sino- Francais de Recherches Avancees (PRA E01 -06: Combustion propre : aspects numeriques etexperimentaux) and the National Natural Science Foundation of China (Grant No. N50106015). The authors alsothank the support from Dr. Kuanfang Ren during the progress of this research.References1. Schuster, A., Radiation through a foggy atmosphere, Astrophysics J., 1905, 21(1): 1一 22.2. Schwarzchild, K, Equilibrium of the Sun' s atmosphere, Nachr. Ges. Wiss. Gottingen Math.-Phys. Klasse,1906,(1):41- 53.3. Maheu, B.. Letoulouzan, J. N, Gouesbet, G., Four-flux models to solve the scattering transfer equation interms of Loren- Mie parameters, Applied Optics, 1984, 23(19): 3353- 3362.4. Maheu, B., Gouesbet, G., Four- flux models to solve the scattering transfer equation: special cases, AppliedOptics, 1986, 25(7): 1122- 1128.5. Roze, C., Girasole, T, Tafforin, A. G.. Multiplayer four flux model of scattering, emitting and absorbing me-dia, Atmospheric Environment, 2001, 35:5125- 5130.6. Modest, M. F, Radiative Heat Transfer, New Y ork: McGraw-Hill Series in Mechanical Engineering, 1993.7. Goodwin, D. G., Mitchner, M.. Flyash radiative properties and effects on radiative heat transfer in coal-firedsystems, International Journal of Heat and Mass Transfer, 1989, 32(4): 627一638.8. Siegel, R., Howell, J. R., Thermal Radiation Heat Transfer, 2nd ed, New York: Hemisphere Publishing Cor-poration, 1980.). Irvine, T. F, Hartnett, J. P., Advances in Heat Transfer, Vol. 3, New York: Academic Press, 1966.10. Roze, C., Girasole, T, Grehan, G. et al, Average crossing parameter and forward scattering ratio values infour- flux model for multiple scattering media, Optics Communication, 2001, 194: 251- 263.11. Wall, T. F., Lowe, A., Wibberley, L. J. et al,, Fly ash characteristics and radiative heat transfer in pulver-ized-coal-fired furnace, Combustion Science and Technology, 1981, 26: 107一121.12. Ozisik, M. N., Radiative Transfer and Interactions with Conduction and Convection, New York: Wiley, 1973.13. Gupta, R. P., Wall, T. F, Truelove,J. s., Radiative scatter by fly ash in pulverized coal-ired furnace: applica-tion of the Monte Carlo method to anisotropic scatter, International Joumnal of Heat and Mass Transfer, 1983,26: 1649- 1660.14. Goodwin, D. G, Infrared optical constants of coal slags, Ph.D Thesis, Stanford University, HTGL ReportT-255, 1986.15. Liu, F S., Swithenbank, J, The effects of particle size distibution and refractive index on fly-ash radiativeproperties using a simplifed approach, International Joural of Heat and Mass Transfer, 1993, 36(7): 1905-1912.16. Grosshandler, W. L, RADCAL: a Narowband Model for I中国煤化工stion Environ-ment, Gaithersburg, MD: National Institute of Standards andYHCNMHGCopyright by Science in China Press 2004