Fundaments of Combustion & Flame

1. Stream lines outer flame temperature Inner flame air premixed flow distance along streamline air air fuel typical burner flame Fundaments of Combustion & Flame diffusion layer reaction layer pre-heated layer fuel oxygen burnt gas flame propagation reaction range unburnt gas diffusion diffusion premixed flame (inner flame) diffusion flame (outer flame)

2. 予熱帯 反応帯 既燃ガス温度 伝播 未燃ガス温度 火炎解析へのアプローチ　–予混合火炎モデル- burnt unburnt Tb ru ub Tu rb uu (燃焼速度) Streamline • 予混合火炎 • 例：ブンゼンバーナ、ガスコンロ • 混合比の定まった予混合ガスの未燃、既燃ガス界面における燃焼 • 火炎は伝播性を持つ • 反応は温度律速 • 火炎特性（火炎面など）を表現する関数 • 反応進行度、火炎伝播速度 ub q uin air fuel

3. 予混合火炎のflameletモデル ● 実用燃焼機器における予混合燃焼流れでは Kolmogrovスケール ≫ 火炎面厚さスケール 流れ変動時間スケール ≫ 化学反応素過程の時間スケール ● G方程式(Kerstein, 1988) ; 火炎面の輸送を表す. SL;(層流)火炎速度 SL δ DT＝Tb－Tu T(t0) T(t1) ruSLCpDT～Shw 　　　　　～rCp(k∂T/∂x)δ

4. Weak points of G-equation modeling • A pure convection equation tends unstable in numerical solution without diffusion term. • An initial profile is conserved in time evolution， even if inappropriate. • Use upwind scheme or add numerical diffusion. • Reset the profile adjusted to physical or mathematical meaning; ex. distance function. 　　　（level-set method） ？？

5. ru ruSu G=G0 Unburnt (G<0) Burnt (0<G) x G 未燃 既燃 d X ， ，・・・ Analysis of local profile near the flame surface d 1D plane flame • Considering a finite thickness d • Add a diffusion term explicitly, • Give a spatial profile of S by Taylor’s expansion around G= G0

6. , Burger’s eq. hyperbolic tangent profile

7. and if Analysis of profile in the flame • Dependency on variation ofr (analogy of ru = const.) • Density weighted flame speed

8. Analysis of profile in the flame (cont.’d) In laminar plane flame ( ) ; • Dependency on Variation of G ( )

9. new 25 steps old 100 steps old+diff. 100 steps １D example 1D Flame propagation by new G-eq. old G-eq. X [mm] X [mm] Fig.1 Time marching solution of new eq. from linear initial profile for CH4:O2:N2=1:2:3 premixed flame. Fig.2 Time marching solutions of new and oldeqs. from the same linear initial profile.

10. ２D example 0.0002[s] • Curvature effect on local flame speed rounds the flame shape. 0.0102[s] • Fluid dynamic flame instability by converging/diverging stream lines. 0.0302[s] Burnt gas flow out 0.0702[s] 0.1102[s] Fig.4 G-profile and steam lines at 0.1802[s]

11. Su r Analysis of solution around a spherical flame 3D formulation • Cylindrical (r-q) coordinate (n: dimension) on flame surface :expand :shrink (extinct) iｆu=0, G=G0,

12. Analysis of solution with a streching flow • Flame in a stretching flow • Flame speed dependency L: Markstain No. (～1) (Calvin 1985) burnt Constant flame speed d x u unburnt

13. 60D Fuel CH4 (99% Vol.) 20D air AIR Coflow D Fuel Inlet diameter D=4.8(mm) Velocity (Fuel ) Ujet=15.0 [m/s] Domain ( X,R,θ)= (60D, 20D, 2π) No. of Grid cells(X,R, θ)=(200,82, 32) Uco=0.74 [m/s] Velocity(Co-flow ) Reynolds No. 4900 Modeling for Partially Premixed Flame Methane-air lifted non-premixed jet flame Muniz and Mungal,Combustion and flame 111, 1997 fuel tube inner diameter D=4.8 [mm] SL0max of methane-air flame=0.37[m] (for ex. Re=4900,avaraged Lift-off height is about 30D=150 [mm]) (www.efluids.com)

14. Air Fuel Blow-out criterion Edge flame extinction Triple flame propagation Frame-front propagation Classification of the flame position Classification of Chen et al.(2000) FLAME C FLAME A

15. 2 scalar flamelet model of partially premixed flame Flamelet equation Diffusion flame, Mixing of Fuel Mixture fraction equation Premixed flame propagation FlameletG-equation Lifted diffusion flame  RANS:Muller et al(1994), Herrmann et al.(2000) LES:Duchamp et al (2001),LES:Hirohata et al(2001) Schematic Figure of Triple flame

16. Air x=xst Burnt gas: Diffusion flame zone Fuel Un-burnt gas: Mixing zone Partial Premixed flame front G=G0 Unburnt mixing zone (G=0) flame surface(G=0.5) Diffusion flame zone (G=1) Mass fraction model using 2scalar flamelet. The G-equation is used to distinguish between the unburnt and burnt regions, the iso-surface of G is used to express flame surface.

17. Quenching effect of turbulent burning velocity L: Markstain No. (～1) Burnt gas with quenching model Burnt gas without quenching model fq=1 Flame tips can not quench where the strong shear exists Lift-off height and flame shape cannot be predicted without quenching model.

18. Premixed flame with the mixture rate gradient Flame speed dependency on defined position based on the flamelet approach G=0.25 G=0.5 G=0.75 18

19. Premixed flame with the mixture rate gradient Fuel ratio (ξ) gradient normal to flame surface (G) Flame speed SL is basically depend on ξ Is flame speed same as simple plane flame? iso-surface of G=0.5 19

20. Premixed flame with the mixture rate gradient mixture rate gradient normal to flame face ⇒ gradient of flame speed ⇒ thinner flame Flame speed gradient on the flame surface : turbulent flame thickness : turbulent flame speed gradient 20

21. G=G(x) x G unburnt burnt d X Level-set to phase field in flame • Distance function (scale in space) • Progress variable (scale in time) plane laminar flame corrugated or wrinkling turbulent flame d d x’: observed distance in averaged flame x : distance from flame surface x (~x’): real distance along streamlimes gas flow d’＞ d：observed thickness (ifs’/d’ ～ s/d)

22. Level-set to phase field in flame If steady solution exists, Steady propagating flame solution: =0 =const =const Level-set form Phase field form if S1→0 (S1/G=const) (F ~ quadratic) thin flame assumption

23. Level-set to phase field in flame Inage’s Hyperbolic Tangent Approximation (Inage et.al. 1989) Example CHEMKIN　（GRI-Mech 3.0） CH4- O2 (φ=1)＋ N2 50%300K h: progress variable Assumed solution for laminar plane flame: temperature CHEMKIN 1400+1100tanh{(x-x0)/d} x ［mm］

24. F（h） h PH for non-equilibrium flame interface (Allen & Cahn 1979, Chen 1992) This formulation insure the second law of thermal mechanics, so that F(h) (=Free energy) decreases in time Allen-Cahn equation Model of steady liquid-solid phase interface (vf=0) 1D plane surface: Model of growing liquid-solid phase interface

25. h 1 0 F(h) PH for non-equilibrium flame interface Gibbs’ free energy progresses in CH4/air flame Functional F(h) of Allen-Cahn eq. q (K) Reaction slow G (kJ/kg) Reaction fast Reaction slow Source term fitting to Inage’s model Gibbs’ free energy: G=H-TS for gas reaction in constant pressure Thermal equilibrium CAN’T be assumed in a flame with large temperature change.

26. Mf(h) h Based on Phase Field Method (Fife 2000) Modified A-C eq. for temperature variation Estimated by numerical solution Inage’s model CHEMKIN solution of homogeneous condition CH4-O2 (stoichimetric 600K) Compare to Inage’s model

27. PH for non-equilibrium flame interface (Fife 2000) h→S： Entropy H(=rcT): Enthalpy is conserved F→G(S,T)：Gibbs’free energy Modified A-C eq. with internal heat sink e Internal heat sink by convection holds a steady flame. S Sb Local homogeneous Su Approx. Tb T ru Tu Local equilibrium assumed in a steady flame (T locally balanced). ⇔ analogy to spinodal phase change x

28. PH for non-equilibrium flame interface Modified A-C eq. for gas reaction with large temperature raise Tb: burnt gas temperature (under H=const. & rcTu ≪DTS) Estimation in flame: Reactions stop at burnt region： M(T) corresponds to the reaction speed which should increase as temperature raise：

29. PH for non-equilibrium flame interface Modified A-C eq. for gas reaction with large temperature raise Tb: burnt gas temperature (under H=const. & rcTu ≪DTS) Ex. CH4/Air premixed flame by num. solution in homogeneous [A] TS[KJ] [D] [B] is estimated by num. solution in homogeneous [C1] [C1] [C2] [C2] and [B] [A] [D] Inage’s model T[K]

30. Conclusive remarks • Modified level-set function for premixed flamelet (G-eq) is derived to consider the flame thickness. • Inage’s flame model (progress variable) is considered by phase-field method based on Allen-Cahn eq. • Modified level-set function is consistent to phase-field method based on Allen-Cahn eq., where the hyperbolic tangent profile is a common approximated solution.