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

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火炎解析へのアプローチ – 予混合火炎モデル - 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 反応帯 既燃ガス温度 未燃ガス温度 伝播 予熱帯 air fuel u in ubub unburnt burnt TuTu Streamline uu u TbTb bb ubub ( 燃焼速度 )

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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 ● 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 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 ● G 方程式 (Kerstein, 1988) ; 火炎面の輸送を表す. S L ;( 層流 ) 火炎速度 SLSL T(t 1 ) T(t 0 ) u S L C p T ～ h ～ C p ∂ T ∂ x)δ T＝Tb－TuT＝Tb－Tu δ

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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 ） ？？

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X G 未燃既燃 Analysis of local profile near the flame surface 1D plane flame Considering a finite thickness –Add a diffusion term explicitly, –Give a spatial profile of S by Taylor’s expansion around G= G 0 x uu uSuuSu Unburnt (G<0) Burnt (0

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Burger’s eq., hyperbolic tangent profile

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Analysis of profile in the flame Dependency on variation of (analogy of u = const.) Density weighted flame speed an d if

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

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

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

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Analysis of solution around a spherical flame 3D formulation Cylindrical ( r - ) coordinate (n: dimension) on flame surface :shrink (extinct) :expand r SuSu i ｆ u=0, G=G 0,

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

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Methane-air lifted non-premixed jet flame Muniz and Mungal,Combustion and flame 111, 1997 fuel tube inner diameter D=4.8 [mm] S L 0 max of methane-air flame=0.37[m] (for ex. Re=4900,avaraged Lift-off height is about 30D=150 [mm]) Velocity(Co-flow ) Fuel Coflow Inlet diameter Reynolds No. AIR D=4.8 (mm) 4900 Velocity (Fuel ) CH4 (99% Vol.) Ujet=15.0 [m/s] Uco=0.74 [m/s] D 60D Fuel air 20D Domain ( X,R,θ)= (60D, 20D, 2π) No. of Grid cells(X,R, θ)=(200,82, 32) (www.efluids.com) Modeling for Partially Premixed Flame

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

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

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

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

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

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

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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

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

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Level-set to phase field in flame Steady propagating flame solution: =const =0=0 if S 1 →0 (S 1 / =const) Level-set form Phase field form (F ~ quadratic) thin flame assumption =const If steady solution exists,

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Level-set to phase field in flame Example CHEMKIN （ GRI-Mech 3.0 ） CH 4 - O 2 (φ=1) ＋ N 2 50% 300K x ［ mm ］ temperature 1400+1100tanh{(x-x 0 )/ } CHEMKIN Inage’s Hyperbolic Tangent Approximation (Inage et.al. 1989) : progress variable Assumed solution for laminar plane flame:

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

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PH for non-equilibrium flame interface 0 1 F Source term fitting to Inage’s model (K) G (kJ/kg) Reaction fast 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. Reaction slow Reaction slow Gibbs’ free energy progresses in CH 4 /air flame Functional F( ) of Allen-Cahn eq.

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Based on Phase Field Method Modified A-C eq. for temperature variation (Fife 2000) Estimated by numerical solution CHEMKIN solution of homogeneous condition CH 4 -O 2 (stoichimetric 600K) Compare to Inage ’ s model Mf Inage ’ s model

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PH for non-equilibrium flame interface Modified A-C eq. with internal heat sink (Fife 2000) → S ： Entropy H(= cT): Enthalpy is conserved F→G(S,T) ： Gibbs’free energy Internal heat sink by convection holds a steady flame. TuTu uu TbTb Local homogeneous Local equilibrium assumed in a steady flame (T locally balanced). ⇔ analogy to spinodal phase change Approx. x T S SuSu SbSb

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

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PH for non-equilibrium flame interface Modified A-C eq. for gas reaction with large temperature raise T b : burnt gas temperature (under H=const. & cT u ≪ TS) [A] by num. solution in homogeneous [B] [C1] is estimated by num. solution in homogeneous [C2] and [D] Inage’s model T [K] TS [KJ] [A] [B] [C2] [C1] [D] Ex. CH 4 /Air premixed flame

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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.

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CHAPTER 8 APPROXIMATE SOLUTIONS THE INTEGRAL METHOD

CHAPTER 8 APPROXIMATE SOLUTIONS THE INTEGRAL METHOD

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