### Slides

```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
• 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: ___?
```