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Report
ME 475/675 Introduction to
Combustion
Lecture 29
Announcements
• Midterm 2
• November 12, 2014 (New Date, not Nov. 5)
• HW 12 (problem 8.2)
• Due Wednesday, November 5, 2014
Tube filled with stationary premixed Oxidizer/Fuel
Products shoot
out as flame
burns in
 Laminar Flame Speed
Burned
Products

Unburned Fuel + Oxidizer
• Video: http://www.youtube.com/watch?v=CjGuHbsi3a8&feature=related (1.49; 3:19; 3:45; 4:05; 4:31; 5:05; 5:35; 6:17; 6:55)
• How to estimate the laminar flame speed  and thickness ?
• Depends on the pressure, fuel, equivalence ratio,…
• Flame reference frame:
 , 
 =  ,  ,
~1 
• 1   +    → 1 +   
• Conservation of mass:  =   =

  ;

=


=
 
 
=
• For hydrocarbon fuels at P = 1 atm,  ≈ 300,  ≈ 2100,  ≈ 7
• What happens within a premixed flame?


Simplified Analysis
Assumptions
Heat and Radical
• One dimensional flow
• Kinetic energy = viscosity = radiation = 0
• Constant pressure
• "
=

−

and

= −



= =
≈


"
• Lewis Number, 
1;


≈ 
• , =  ≠    
• Single Step Kinetics
• Φ < 1, Fuel Lean, so fuel is completely consumed
Conservations Laws
• Mass Conservation
"
•  =  = ;
 


=0
• Species Conservation
•
"
•
′′′
+  
=
 "

=
′′′


=
"
"
"  
 +


"  − 



(using Flick’s Law)
• Apply to: 1   +    → 1 +  ; Air/Fuel ratio  =
1

′′′
• ′′′ = 
=−
1
′′′

+1
•  = 1, 2, … ,  = 3; Fuel, Oxidizer, Products
•
•
•







  − 


"
  − 


"
  − 

"
′′′
= ′′′
′′′
= 
=  ′′′
′′′
= 
= − 1 +  ′′′
 
 
"



 " +


Energy Conservation (Ch. 7, pp. 239-244)

′′ ℎ
′′

•  −  =  ℎ − ℎ
•
•
′′
 ′′
−

−
′′
=
ℎ
′′


+
 ′′


= 
′′
ℎ+
ℎ


−ℎ
• Decreasing heat flux in x-direction increases enthalpy in the +x-direction
′′
ℎ
ℎ+


′′



 ′′ +


Heat Flux
• Heat: Energy transfer at a boundary due to temperature difference
• When there is a large species gradient, diffusion contributes to heat flux
•
′′
=

−

• Note:
•
′′
ℎ

=
+
′′
,
ℎ


 ℎ =
•

ℎ
 
=  −  
ℎ

=

−

− 
ℎ

• Heat Flux =

ℎ
 
−




−

=
+

=
ℎ

=

−

−


ℎ
 

ℎ
 
+
ℎ
− 

=
 ,
+

−



=



• Flux due to conduction +
• Flux of standardized enthalpy due to species diffusion +
• Flux of sensible enthalpy due to species diffusion
• For  =


=


≈ 1;  ≈ 
• Shvab-Zeldovich assumption ( ≈ (1) for most combustion gases)
ℎ
′′
•  = −
(due to both conduction and diffusion)

− 

ℎ
 
+ 

ℎ
 


Shvab-Zeldovich form of Energy Conservation
•
 ′′
−

•ℎ=
•
•
′′
•
•
•
•




ℎ

=
=
ℎ
′′



 ℎ,
 ℎ =
=
 
ℎ
 ,
ℎ
−

 ′′
−

=


(Energy Equation)
+

ℎ,
 

+
 

  
+
= −
 
ℎ
 ,

 

ℎ,

 

ℎ
 ,


−
ℎ,
′′ 


  , 

+
−
−
"



 
ℎ
 ,
+
+



=
=

 ℎ,
 
ℎ
 ,
+
+

 
 






 

=
ℎ, + 






′′
′′
  ℎ, =   −








′′

=   −





′′
Shvab-Zeldovich form of Energy Conservation
•

−


ℎ,
•

−


ℎ,
" =



′′
− 
=   −



"
"





− ℎ,
= − ℎ,
′′′

′′ 
• Species conservation:
•
 "




= ′′′




′′′
′′′
ℎ,
′′′ = ℎ,
′′′ + ℎ,

+ ℎ,

•



= ℎ,
′′′ + ℎ,
′′′ − ℎ,
1 +  ′′′
•



= ′′′ ℎ,
+ ℎ,
 − ℎ,
1+
= ′′′ Δℎ



• Δℎ = ℎ,
+ ℎ,
 − ℎ,
1 +  : Heat of combustion
• For  =


=


≈ 1;  ≈ 
• −′′′ Δℎ = ′′ 


−





nd
2
order differential
equation for T(x)
•
′′ 


−
1 
 



• Where ′′ =  
=
′′′

Δℎ
−

• Only accepts 2 boundary conditions,
• But we have 4 (Eigenvalue problem)
• For  → −∞:  →  
• For  → +∞:  →  




→0
→0
• For an approximate solution, assume a simple profile
• Find flame thickness  and laminar flame speed  =
conditions can be satisfied
′′

so that all four boundary
Approximate Solution
•

′′


−
1 
 
• Integrate
•
′′



−



+∞
−∞
1

=
′′′

Δℎ
−


 0

 0
=
Δℎ +∞ ′′′
−
 
 −∞
• ′′′ =   = 0   < 0   > 
• Inside 0 <  < ,
•
′′
•
′′


=
 −
,

so  =
 −  =
Δℎ +∞ ′′′
−

 −∞
 −  =
Δℎ  ′′′
−





 −
eqn. 1
• Two unknowns: ′′ =   and 
• Need another equation


 −
=
 ′′′
Δℎ  1
−

  − 
 
Average over
temperature: ′′′
Approximate Solution
•

′′


−
1 
 
• Integrate
•
•
•
•



/2
−∞
=
′′′

Δℎ
−


 −

 +
2

 
Δℎ /2 ′′′
 
−
=−
 
  0
 −∞
 +
  −
′′   −  −
=0
2


′′
 −
  −
′′

=
2


2
 = ′′ eqn. 2
 
• From eqn. 1
• ′′ =
′′
 −  =
2Δℎ
2  −
Δℎ  ′′′
−


−′′′ =   ;  =
1

=
≈0
Δℎ ′′′ 2
−
 ′′

 
2Δℎ
2  −
−′′′
Approximate Solution
•  =
1

2Δℎ
2  −
−′′′
• But Δℎ = 1 +    − 
• Show this in HW (problem 8.2)
•  =
•
•
2 1+   −
  2  −
−′′′
2 1+
 =
−′′′

2
2
eqn. 2:  = ′′ =
 
  
•=
2
′′′ 1+
− 
=
2

=
; where  =
2

=

 
2
2 1+

′′′
− 
(Fast flames are thin)
Example 8.2
• Estimate the laminar flame speed of a stoichiometric propane-air mixture using
the simplified theory results (Eqn. 8.20). Use the global one-step reaction
mechanism (Eqn. 5.2, Table 5.1, pp. 156-7) to estimate the mean reaction rate.
• Find: ___?

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