Non-Premixed Laminar Flames Introduction • Design of burners, in general, relies heavily on “art and craft” of burner design • After stability, the next main concern is emission • NOx and CO are toxic gases • Flame geometry, particularly compactness, is a primary concern • Fuel type must be considered • Methane vs propane grill, for example Non - reacting laminar fuel jet flowing into an infinite oxidizer reservoir Uniform jet velocity profile Free shear layer: fuel and air mix by molecular diffusion Oxidizer is "entrained" as jet momentum is transferred to it Pure fuel inside, pure oxidizer outside Potential core: region free from viscous effects, no mixing Initial jet momentum and mass conserved throughout the flow field ∞ 2π ∫ ρ ( r , x ) v x2 ( r , x ) rdr = ρev e2 π R 2 0 ∞ 2π ∫ ρ ( r , x ) v x ( r , x )YF ( r , x ) rdr = ρev eπ R 2YF ,e 0 Assumptions Jet and oxidizer MW's are equal, Ideal gas, P, T, and ρ are constant Fick's law diffusion Equal momentum and species diffusivities, = = ν= 1 ν D, Sc D Neglect axial diffusion, radial only Boundary conditions along the jet centerline, r = 0 v r ( 0, x ) = 0 Mass ∂v x 1 ∂ (v r r ) + = 0 ∂x r ∂r Momentum ∂v x (0, x ) = 0 ∂r ∂YF (0, x ) = 0 ∂r vx far from jet, r → ∞ ∂YF ∂YF 1 ∂  ∂YF  + vr = D r ∂x ∂r r ∂r  ∂r  Yox = 1 − YF v x ( ∞, x ) = 0 YF ( ∞, x ) = 0 at jet exit, x = 0 v x ( r ≤ R, 0 ) = ve v x ( r > R, 0 ) = 0 YF ( r ≤ R, 0 ) =YF ,e =1 YF ( r > R, 0 ) = 0 ∂v x ∂v 1 ∂  ∂v x  + vr x = ν r ∂x ∂r r ∂r  ∂r  Species vx Note: for ν = D species and momentum have the same functional solution form Solution Similarity solution: profiles are "similar", i.e., intrinsic shape of velocity profiles are the same everywhere Profiles depend only on the similarity variable r x vx 3 Je  ξ 2  1 +  8π µ x  4  2 QF = v eπ R 2  3ρ J  ξ = e e  16π  1 ( ) ( ) along the centerline, r = 0 → ξ = 0 2 3 QF  ξ  YF = 1 +  8π Dx  4  3   ξ 1 ξ−  2 4   3Je  1  vr =   16 x  ξ 2 2 πρ e   1 + 4    Je = ρev e2 π R 2 2 ρ v R 1 1 = 0.375  e e  2 ve  µ  x  ξ2  R 1 + r   ρ v R 1 1 YF = 0.375  e e  2  µ  x  ξ2  R 1 + r   vx vx ve = 0.375 YF ,0 = 0.375 Re j (x R) Re j (x R) ∴ centerline velocity decays with 1 and increases with Re j Solutions valid far from jet, i.e., 2 r µx ( x R ) > 0.375 Re j x Jet half - width, spreading rate and angle Jet half-width: jet velocity is half the centerline value vx v x ,0 =1 r1 2 x 2 and solve for r = 2.97 Re j  r1 2 α = tan−1    x  Jet flame physical description Flame surface is the locus of points where φ = 1 Overventilated: excess O2 in surroundings Underventilated: deficient O2 in surroundings Laminar non - premixed flame length Flame length depends on QF but not on v e or R independently Lf ≈ QF 3 8π DYF ,stoic Buoyancy tends to accelerate and narrow flame increasing concentration gradient and diffusion. The two effects tend to cancel so that simple theories neglecting buoyancy well-predict Lf Simplified theory Burke-Schumann (1928 ) , and Roper (1977 ) Assumptions 1. Laminar, steady flow, radial port of radius R 2. Quiescent, infinite reservour of oxidizer 3. Species: fuel, oxidizer, products 4. Fuel and oxidizer react at φ = 1 5. Fast chemical kinetics (flame sheet approximation) 6. Fick's law diffusion 7. Le= 1= α D 8. Neglect thermal radiation and axial diffusion 9. Vertically oriented flame Mass conservation 1 ∂ ( r ρv r ) ∂ ( ρv x ) + = 0 ∂r ∂x r Axial momentum 1 ∂ ( r ρv x v x ) 1 ∂ ( r ρv x v r ) 1 ∂ + − r r r ∂r ∂x ∂r  ∂v x  ( ρ∞ − ρ ) g  r µ ∂r  =   Species conservation 1 ∂ ( r ρv xYi ) 1 ∂ ( r ρv r Yi ) 1 ∂ + − r r r ∂r ∂x ∂r ∂Yi   r D 0 = ρ   ∂r   YPr =1 − YF − YOx Energy conservation ( ) ( ) ∂ ∂ ∂ r ρv x ∫ c p dT + r ρv r ∫ c p dT − ∂r ∂r ∂x  ∂ c dT  r ρD ∫ p  ∂r   = 0   Unknowns: v r , v x , T , YF , and YOx Conserved scalar approach requires BCs only along flame sheet, far from jet, and jet exit A conserved scalar is any scalar quantity that is conserved throughout the flow field mass of material having it's f ≡ origin in the fuel stream mass of mixture 1 kg fuel + ν kg oxidizer → (ν + 1 ) kg products f = 1) ( YF  kg fuel stuff kg fuel kg mixture kg fuel  1  YPr + ( 0 ) YOx +   ν + 1  kg     product kg fuel stuff kg oxidizer kg fuel stuff kg mixture kg product 1 Y = YF + fstYPr ν + 1 Pr f = YF + kg oxidizer kg mixture Species conservation, conserved scalar approach Replace two species conservation equations with one involving the mixture fraction f , which is the fraction of material having its origin in the fuel system 1 ∂ ( r ρv x f ) 1 ∂ ( r ρv r f ) 1 ∂ + − r r r ∂r ∂x ∂r ∂f (0, x ) = 0 by symmetry ∂r f ( ∞, x ) = 0 f ( r ≤ R, 0 ) = 1 f ( r > R, 0 ) = 0 Flame situates itself at f = fstoic ∂f  r D ρ  ∂r   0 =  Energy conservation, conserved scalar approach 1 ∂ ( r ρv x h ) 1 ∂ ( r ρv r h ) 1 ∂ + − r r r ∂r ∂x ∂r ∂h   r D 0 ρ =   ∂r   ∂h (0, x ) = 0 by symmetry ∂r h ( ∞, x ) = hOx h ( r ≤ R, 0 ) = hF h ( r > R, 0 ) = hOx Continuity and axial momentum No change in equations or BCs Non - dimensional forms x* = h − hOx ,∞ vx vr x r ρ r* v x* v r* h* ρ* = = = = = ρe R R ve ve hF ,e − hOx ,∞ Continuity ( ∂ ρ *v x* ) + 1 ∂ (r * ρ *v x* ) r* ∂x * ∂r * Axial momentum ( ∂ r * ρ *v x* v x* ∂x * ) + ∂ (r * 0 = ρ *v x* v r* ) ∂r * ∂ − * ∂r  µ  * ∂v x*  gR  ρ∞ *  * r = − ρ   r *  2   ρev e R  ∂r  v e  ρe  Mixture fraction ( ∂ r * ρ *v x* f ∂x * ) + ∂ (r * ρ *v r* f ) ∂r * − ∂ ∂r *  ρ D  * ∂f  0 =  r *   ρev e R  ∂r  Energy ( ∂ r * ρ *v x* h * ∂x * ) + ∂ (r * ρ *v r* h * ) ∂r * ∂ − * ∂r  ρ D  * ∂h *  0 =  r *  v R ρ r ∂  e e   Boundary conditions ( ) ∂f , x ∞ = ( ) ∂r ( ∞, x ) =h ( ∞, x ) =0 v r* 0, x * = 0 v x* * * * * ∂v x* ∂f ∂h * * * 0, x = 0, x = ( ∞, x )= 0 ∂r * ∂r * ∂r * ( ( (r ) ( ) ( > 1, 0 ) = f ( r ) ) ( > 1, 0 ) = h ( r ) > 1, 0 ) = 0 v x* r * ≤ 1, 0 = f r * ≤ 1, 0 = h * r * ≤ 1, 0 = 1 v x* * * * * • Mixture fraction and standardized enthalpy equations are of the same form, as are their boundary conditions. ∴ their solutions are identical. µ ν 1 and neglecting buoyancy yields • Assuming Sc = = = ρD D for axial momentum, mixture fraction, and enthalpy ( ∂ r * ρ *v x* ζ ∂x * ) + ∂ (r * ρ *v r*ζ ) ∂r * − ∂  1  * ∂ζ  0 r  = ∂r *  Re  ∂r *  ζ = f= v x* = h * • ρ * and v x* are coupled through continuity • ρ * and f (or h * ) are coupled through state relations State relations Need to relate density to mixture fraction or other conserved scalars = YF YF ( = f) YPr YPr= ( f ) YOx YOx= ( f ) T T= (f ) ρ ρ (f ) Inside flame sheet ( fst < f ≤ 1) = YF f − fst 1−f = YOx 0= YPr 1 − fst 1 − fst At flame = YF 0 = YOx 0= YPr 1 Outside flame f YF == 0 YOx 1 − f YPr = fst fst Note: fst = 1 ν +1 Temperature state relation Assume: c= c p= c= cp ,Ox p,Pr p,F hf0,P = hf0,Ox = 0 and hf0,F = ∆hc Enthalpy h= ∑Y h i i = YF ∆hc + c p (T − Tref ) YF ∆hc + c p (T − TOx ,∞ ) h − hOx ,∞ h = = = f hF ,e − hOx ,∞ ∆hc + c p (TF ,e − TOx ,∞ ) * c p (TOx ,∞ − Tref ) and hF ,e = ∆hc + c p (TF ,e − Tref ) using hOx ,∞ = Solve for T T =− ( f YF ) ∆hc + f (TF ,e − TOx ,∞ ) + TOx ,∞ cp Substituting appropriate expressions for YF inside, outside, and on the flame sheet Inside ( fst < f ≤ 1)  fst ∆hc  T =T ( f ) =TF ,e f + TOx ,∞ +  (1 − f ) − 1 f c  st p   At the flame ( f = fst )  ∆hc  + TF ,e − TOx ,∞  + TOx ,∞ T = T ( f ) = fst   c   p  Outside the flame ( 0 ≤ f < fst )  ∆h  T = T ( f ) = f  c + TF ,e − TOx ,∞  + TOx ,∞  c   p  Roper Constant density solution Lf ≈ 3 QF 8π DYF ,st Variable - density approximate solution µ = µref T Tref F m ρ∞ 3 1 8π YF ,st µref ρref I  ρ∞   ρ   f  ρ I  ∞  is a momentum integral tabulated in Table 9.2  ρf  The variable density solution predicts lengths about 2.4 times longer than constant density solutions. Lf ≈ All theories predict that flame lengths depend on Volume flow rate and 1 YF ,st independent of port diameter Circular port Lf ,th QF    TF  = 4π D∞ ln 1 + 1 T∞ ( S ) T∞   T   F  0.67 T QF  ∞  TF   Lf ,exp = 1330 ln 1 + 1 S where S is the molar oxidizer-fuel ratio Square port ( ) T QF  ∞   TF  Lf ,th = ( 16D∞ Inverf 1 + 1 Lf ,exp = 1045 S ) T QF  ∞   TF  (Inverf Slot burners see text for expressions 1+ 1 S 2 ) T∞   T   F  2 0.67 S= 1 − ψ pri ψ pri + 1 Spure ψ pri is the ratio of primary air to the amount required at stoichiometric S= x+y χO 2 4 Diluent effect S= χ dil x+y 4  1    χO2 1 − χ dil   is the diluent mole fraction in the fuel stream Study Turns Example 9.4 • How does flame height change for increase/decrease of primary aeration? • How does flame height change for increased jet flow rate? • How does replacing methane with propane affect the burner design assuming the total burner power remains the same? • What is the effect of aeration with oxygen rather than air? Soot formation
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