# Combustion questions laminar non premixed flames

Anonymous
timer Asked: Dec 10th, 2017
account_balance_wallet \$25

Question description

My professor added one section that will be covered on the conceptual part of the exam. Could you make a quick study guide for this section? Specifically explaining the graphs/flame diagram (slides 18, 21, 22, 23, 24) and answering the questions on slide 25. Anything else you can think of would be appreciated too.

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

## Tutor Answer

Aljon2017
School: UIUC

Hi! Please see the following file:The first part provides answers to questions on slide 25.The next part provides an additional information.The later parts are explanation of the graphs that are mentioned.If you will notice later on, I did not discuss all graphs deliberately on the last part because some of them are already mentioned on the first/earlier parts but I want to assure you that everything is covered :)

ANSWER TO QUESTIONS in Page 25:
1. How does flame height change for increase/decrease of primary aeration?

2. How does flame height change for increased jet flow rate?

They are directly proportional.

3. How does replacing methane with propane affect...

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Review

Anonymous
Outstanding Job!!!!

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