Transcript Slides
ME 475/675 Introduction to
Combustion
Lecture 39
Announcements
• HW16 Ch. 9 (8, 10,12)
• Due Wednesday, 12/2/2015
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Flame length (a measurable quantity)
• For un-reacting fuel jet (no buoyancy)
3
• 𝐿𝐹 = 8 𝑅𝑒𝑗 𝑌
𝑅
𝐹,𝑠𝑡
3
𝑄
3
𝑄
= 8𝜋 𝜈𝑌 𝐹 = 8𝜋 𝒟𝑌 𝐹
𝐹,𝑠𝑡
• Jet Reynold number: 𝑅𝑒𝑗 =
𝜌𝑒 𝑣𝑒 𝑅
𝜇
𝐹,𝑠𝑡
𝑣𝑒 𝑅
=
𝜈
=
𝑣𝑒 𝑅
𝒟
• Experimental Roper Correlations include buoayancy pp. 336-9
𝑄𝐹 𝑇∞ 𝑇𝐹
~ 𝑄𝐹
ln(1+1 𝑆)
𝑄 𝑇∞ 𝑇𝐹
1045 𝑖𝑛𝑣𝑒𝑟𝑓𝐹 1+𝑆
−0.5 2 ~ 𝑄𝐹
• round nozzles: 𝐿𝑓,𝑒𝑥𝑝 = 1330
• square nozzles: 𝐿𝑓,𝑒𝑥𝑝 =
• Slot Nozzle in stagnant oxidizer are dependent on Froude number
• 𝐹𝑟𝑓 =
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑗𝑒𝑡 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦
=
𝑣𝑒 𝐼𝑌𝐹,𝑠𝑡𝑖𝑜𝑐
2
=
𝑎𝐿𝑓
𝑣𝑒 𝐼𝑌𝐹,𝑠𝑡𝑖𝑜𝑐
0.6𝑔
1500𝐾
𝑇∞
2
−1 𝐿𝑓
• Plug nozzle velocity profile: 𝐼 = 1; Parabolic profile : 𝐼 = 1.5;
Initially guess = 1 m
𝑇∞ 2
~ 𝑄𝐹 ;
𝑓,𝑆𝑡𝑜𝑖𝑐 𝑇𝐹
𝑏𝛽2 𝑄𝐹
• 𝐹𝑟𝑓 ≫ 1: Momentum Controlled: 𝐿𝑓,𝑒𝑥𝑝 = 86,000 𝐻𝐼𝑌
• 𝐹𝑟𝑓 ≪ 1: Buoyancy Controlled: 𝐿𝑓,𝑒𝑥𝑝 = 2000
4
• 𝐹𝑟𝑓 ~ 1: Transitional: 𝐿𝑓 = 9 𝐿𝑓,𝑀
𝐿𝑓,𝐵
𝐿𝑓,𝑀
𝑄𝐹 𝛽𝑇∞
ℎ𝑇𝐹
3
1 + 3.38
4
1 3
1
𝑎
𝐿𝑓,𝑀
𝐿𝑓,𝐵
4
~ 𝑄𝐹 3
3 2 3
−1
𝛽=
1
1
4 𝑖𝑛𝑣𝑒𝑟𝑓 1+𝑆
Effect of Oxidizer Oxygen Content (𝜒𝑂2 < or > 21%)
• Circular tube
• 𝐿𝑓,𝑒𝑥𝑝 =
𝑄𝐹 𝑇∞ 𝑇𝐹
1330
ln(1+1 𝑆)
=
𝑄𝐹 𝑇∞ 𝑇𝐹
1330
ln 1+𝜒𝑂2 2
• Stoichiometric moles of oxidizer per mole of fuel
• 𝑆=
1
𝜒𝑂2
𝑥+
𝑦
4
• For methane 𝑆𝐶𝐻4 =
•
𝐿𝑓,𝜒𝑂
2
𝜒𝑂2
𝑄 𝑇
𝑇
1330 𝐹 ∞ 𝐹
2
𝐿𝑓,𝜒𝑂 =21%
2
=
ln 1+𝜒𝑂 2
2
𝑄 𝑇
𝑇𝐹
1330ln𝐹1+∞
0.21 2
=
ln(1+0.21 2)
ln 1+𝜒𝑂2 2
• Increasing 𝜒𝑂2 in oxidizer decreases flame
length
• Circular tube flame length
Fuel Dependence
• 𝐿𝑓,𝑒𝑥𝑝 = 1330
(increases with S)
• For stoichiometric reaction of a generic HC fuel
• 𝐶𝑥 𝐻𝑦 + 𝑥 +
• 𝑆=
𝐶4 𝐻10
𝑄𝐹 𝑇∞ 𝑇𝐹
ln(1+1 𝑆)
𝑁𝑂𝑥
𝑁𝐹𝑢 𝑆𝑡
𝑦
4
=
𝑂2 + 3.76𝑁2 → ⋯
1
𝜒𝑂2
𝑁𝑂2
𝑁𝐶𝑥 𝐻 𝑦
=
𝑆𝑡
1
𝜒𝑂2
𝑥+
𝑦
4
𝑦
• If air is the oxidizer, then 𝜒𝑂2 = 0.21 and 𝑆 = 4.76 𝑥 +
4
𝑦
• If the oxidizer is pure 𝑂2 , then 𝜒𝑂2 = 1 and 𝑆 = 𝑥 +
4
𝐶3 𝐻8
• Alkane Fuels: 𝐶𝐻4 , 𝐶2 𝐻6 , 𝐶3 𝐻8 ,… 𝐶𝑥 𝐻2(𝑥+1) , y = 2(𝑥 + 1)
• 𝑆 = 4.76 𝑥 +
2(𝑥+1)
4
= 4.76 1.5𝑥 + 0.5
• 𝑆𝐶𝐻4 = 9.52
𝐶2 𝐻6
• For a given flow rate 𝑄𝐹 and air oxidizer
•
𝐶𝐻4
𝐿𝑓
𝐿𝑓,𝐶𝐻4
=
ln(1+1 𝑆𝐶𝐻4 )
ln(1+1 𝑆)
=
ln(1.11)
ln 1+
1
4.76 1.5𝑥+0.5
,
• Heavier fuels require more air, and so more time and distance (𝐿𝑓 )
to reach the stoichiometric condition
𝐻2 𝑜𝑟 𝐶𝑂
• Light fuels, 𝑆 =
1
2
1
+
2
4.76
2
= 2.33 (short flame)
• 𝐻2 +
𝑂2 + 3.76𝑁2 → 𝐻2 𝑂 + 1.88𝑁2
• 𝐶𝑂
𝑂2 + 3.76𝑁2 → 𝐶𝑂2 + 1.88𝑁2
Stoichiometric Factors
• When nozzle gas is pure 𝐶𝑥 𝐻𝑦 fuel and ambient gas is air with 𝑂2 mole fraction 𝜒𝑂2
• 𝑆=
1
𝜒𝑂2
𝑁𝑂2
𝑁𝐶𝑥 𝐻𝑦
=
𝑆𝑡
1
𝜒𝑂2
𝑥+
𝑦
4
= 𝑆𝑃𝑢𝑟𝑒 (pure fuel)
• Now, generalize the Roper Correlations to include:
• Air or inter gas added to fuel (fuel is “aerated” or “diluted”)
• Revise the definition of 𝑆 =
𝑁𝐴𝑚𝑏𝑖𝑒𝑛𝑡
𝑁𝑁𝑜𝑧𝑧𝑒𝑙 𝑆𝑡
• Number of moles of the ambient gas per mole of nozzle gas when fuel to O2 ratio is stoichiometric.
• Adding air to fuel makes it more premixed and shortens the flame length
• Less oxidizer needs to diffuse in to reach stoichiometric conditions
• Flame is bluer and less sooty; Keep A/F ratio low enough so that equivalence ratio Φ =
A/F 𝑠𝑡 A/F is above rich limit to avoid flashback
• Adding inert gas (such as 𝑁2 or products of combustion) to the fuel reduces flame
temperature and oxides of nitrogen, and increases flame length
• More moles of oxidizer need to diffuse in per mole of nozzle gas to reach stoichiometric conditions
Primary Aeration (of nozzle)
•
𝑁𝐴𝑚𝑏
Ambient
= 1 − 𝜓𝑃𝑟𝑖 𝑆𝑃𝑢𝑟𝑒
•
𝑁𝐴𝑚𝑏𝑖𝑒𝑛𝑡
1−𝜓𝑃𝑟𝑖 𝑆𝑃𝑢𝑟𝑒 1 𝑆𝑃𝑢𝑟𝑒
𝑆=
=
𝑁𝑁𝑜𝑧𝑧𝑒𝑙 𝑆𝑡
1+𝜓𝑃𝑟𝑖 𝑆𝑃𝑢𝑟𝑒 1 𝑆𝑃𝑢𝑟𝑒
1−𝜓𝑃𝑟𝑖
=
1 𝑆𝑃𝑢𝑟𝑒 +𝜓𝑃𝑟𝑖
𝐿𝑓
ln(1+1 𝑆𝑃𝑢𝑟𝑒 )
ln(1+1 𝑆𝑃𝑢𝑟𝑒 )
=
=
1 𝑆
+𝜓
𝐿𝑓,𝑝𝑢𝑟𝑒
ln 1+1 𝑆
ln 1+ 𝑃𝑢𝑟𝑒 𝑃𝑟𝑖
1−𝜓𝑃𝑟𝑖
• For methane 𝑆𝑃𝑢𝑟𝑒 = 9.52
Fuel+Oxidizer
𝑦
𝐶𝑥 𝐻𝑦 + 𝜓𝑃𝑟𝑖 𝑥 +
𝑂2 + 3.76𝑁2
4
𝜓𝑃𝑟𝑖
𝑦
𝑁𝑁𝑜𝑧 = 1 +
𝑥+
= 1 + 𝜓𝑃𝑟𝑖 𝑆𝑃𝑢𝑟𝑒
𝜒𝑂2
4
1
Fuel Dilution
•𝑆=
Ambient
𝑁𝐴𝑚𝑏 = 𝑆𝑃𝑢𝑟𝑒
𝑦
𝑥+4
=
𝜒𝑂2
•
•
Fuel+Oxidizer
𝐶𝑥 𝐻𝑦 + 𝑁𝐷 𝐷
𝑁𝐴𝑚𝑏𝑖𝑒𝑛𝑡
𝑁𝑁𝑜𝑧𝑧𝑒𝑙 𝑆𝑡
=
𝑦
𝜒𝑂
2
𝑥+ 4
1+𝑁𝐷
𝑦
=
𝑥+ 4
𝜒𝑂2 1+𝑁𝐷
𝑁𝐷
1
1
1
1
• 𝜒𝐷 =
=
;
=
+ 1; 𝑁𝐷 =
1+𝑁𝐷
1 𝑁𝐷 +1 𝜒𝐷
𝑁𝐷
1 𝜒𝐷 −1
𝜒𝐷
• 𝑁𝐷 =
1−𝜒𝐷
𝑦
𝑦
𝑦
𝑥+ 1−𝜒𝐷
𝑥+ 4
𝑥+ 4
4
𝑆=
=
=
𝜒𝐷
1
𝜒𝑂2
𝜒𝑂2 1+
𝜒𝑂2
1−𝜒𝐷
1−𝜒𝐷
𝐿𝑓
𝐿𝑓,𝑝𝑢𝑟𝑒
=
ln(1+1 𝑆𝑃𝑢𝑟𝑒 )
ln 1+1 𝑆
=
ln(1+1 𝑆𝑃𝑢𝑟𝑒 )
ln 1+
𝜒𝑂
2
𝑦
𝑥+ 4 1−𝜒𝐷
𝑦
4
• For methane 𝑆𝑃𝑢𝑟𝑒 = 9.52, 𝑥 + = 2
• For air 𝜒𝑂2 = 0.21
•
𝐿𝑓
𝐿𝑓,𝑝𝑢𝑟𝑒
=
ln(1.105)
ln 1+
0.105
1−𝜒𝐷
Example 9.4
• Design a natural-gas burner for a commercial cooking range that has a
number of circular ports arranged in a circle. The circle diameter is
constrained to be 160 mm (6.3 inch). The burner must deliver 2.2 kW
at full load and operate with 40 percent primary aeration. For stable
operation, the loading of than individual port should not exceed 10 W
per mm2 of port area. (see Fig. 8.25 for typical design constraints for
natural-gas burners.) Also, the full-load flame height should not
exceed 20 mm. Determine the number and the diameter of the
ports.
Ch. 10 Droplet Evaporation and Burning
• Liquid Fuels → High Pressure Atomizer → Droplets → Evaporation →
Non-pre-mixed flame
• Spray Combustion Applications (more complex than droplet burning)
• https://www.youtube.com/watch?v=a6Z_SpU1MQs&feature=channel&list=UL
• Applications: Spray Combustion (not droplet burning)
• Oil home heaters (http://www.oilheatamerica.com/index.mv?screen=burners)
Applications: Diesel Engines
Injector
• Diesel Fuels: Less volatile
(prone to evaporate) than
spark-ignition fuels but
more easily auto-ignited
(at high pressures and
temperatures)
• Engines
• Indirect injection
• Direct injection
Pre-mix
chamber
Glow Plug
• Droplets evaporate and
premix with air, burn then
auto-ignite the rest of the
mixture
• Blows into main chamber
and completes combustion
Gas Turbine
Engines
(aircraft and
stationary)
• Annular Combustor is a relatively small component
Annular Multistage Combustor
• Fuel is atomize
• Premixed and staged to avoid NOx formation
• Walls are protected from high temperatures by film cooling
Liquid Rocket Engines (fuel and oxidizer are liquid)
• Pressure-fed by high
pressure gas
• Pump-fed by turbopumps
• Mixed by colliding jets
to form unstable
sheets and break up
Flame length (a measurable quantity)
• Flame length 𝐿𝑓 :
• Φ 𝑟 = 0, 𝑥 = 𝐿𝑓 = 1; 𝑌𝐹 = 𝑌𝐹,𝑠𝑡
• For un-reacting fuel jet (no buoyancy)
• For Schmidt number 𝑆𝑐 =
𝜈
𝒟
= 1,
𝑅
𝑌𝐹 = 0.375𝑅𝑒𝑗 𝑥
3𝜌 𝐽 1 2 1 𝑟
• Dimensionless Similarity Variable: 𝜉 =
• Jet Reynold number: 𝑅𝑒𝑗 =
𝜌𝑒 𝑣𝑒 𝑅
𝜇
=
𝑣𝑒 𝑅
𝜈
1+
𝜉2
4
−2
•
•
=
𝑒 𝑒
16𝜋
𝑣 𝑅
= 𝑒
𝒟
𝜇𝑥
Y( x y)
3 𝜌𝑒 𝑄𝐹
8𝜋 𝜇𝑌𝐹,𝑠𝑡
=
3 𝑚𝐹
8𝜋 𝜇𝑌𝐹,𝑠𝑡
=
• Decreases with increasing 𝒟 and 𝑌𝐹,𝑠𝑡 =
1
𝑚
1+ 𝑂𝑥
𝑚𝐹𝑢
=
3 𝑄𝐹
8𝜋 𝜈𝑌𝐹,𝑠𝑡
=
3 𝑄𝐹
8𝜋 𝒟𝑌𝐹,𝑠𝑡
0.095
1
𝑁 𝑀𝑊
1+𝑁𝑂𝑥 𝑀𝑊𝑂𝑥
𝐹𝑢
𝐹𝑢
=
1
𝑀𝑊
1+𝑆𝑀𝑊𝑂𝑥
𝐹𝑢
;
0.06
Y ( x Ya ( x) )
X( x Ya ( x) )
0.04
𝑦
4
,
𝑀𝑊𝑂𝑥
𝑀𝑊𝐹𝑢
=
28.85
12.011𝑥+1.00794𝑦
0.02
• For y = 2x+2 (alkanes), decreases with increasing x
• What is the effect of buoyancy?
1
y
28.85
1 4.76 x
4 12.011 x 1.00794y
0.08
• Depend on fuel
• For 𝐶𝑥 𝐻𝑦 fuel, 𝑆 = 4.76 𝑥 +
y
4
Ya ( x) ( 2 x) 2
• Increases with 𝑄𝐹 = 𝑣𝑒 𝜋𝑅2 (not dependent on 𝑣𝑒 𝑜𝑟 𝑅 separately)
• What about 𝒟?
1 4.76 x
• Flame length, x = 𝐿𝐹 where 𝑌𝐹 = 𝑌𝐹,𝑠𝑡 at 𝑟 = 𝜉 = 0
−2
𝑅
02
𝑌𝐹,𝑠𝑡 = 0.375𝑅𝑒𝑗
1+
𝐿𝐹
4
3
𝑅
3 𝜌𝑒 𝑣𝑒 𝑅
𝑅𝜋
𝐿𝐹 = 𝑅𝑒𝑗
=
8
𝑌𝐹,𝑠𝑡
8
𝜇
𝑌𝐹,𝑠𝑡 𝜋
1
X( x y)
0
0
2
1
4
6
x
8
10
11
Experimentally-Confirmed Numerical Solutions
• Roper Correlations pp. 336-9; Table
9.3, Equations 9.59 to 9.70
• Subscripts:
• thy = Theoretical
• expt = Experimental
• Experimental results
• round nozzles,
• 𝐿𝑓,𝑒𝑥𝑝 = 1330
𝑄𝐹 𝑇∞ 𝑇𝐹
ln(1+1 𝑆)
~𝑄𝐹
• square nozzles,
• Inverse Gaussian
error function
“inverf” from
Table 9.4
• 𝐿𝑓,𝑒𝑥𝑝 = 1045
𝑄𝐹 𝑇∞ 𝑇𝐹
~𝑄𝐹
𝑖𝑛𝑣𝑒𝑟𝑓 1+𝑆 −0.5 2
• Metric units (m, m3/s)
• S = Molar Stoichiometric ratio =
4.75*(x+y/4) for CxHy fuel
• Temperatures: 𝑇∞ oxidizer, 𝑇𝐹
Fuel, 𝑇𝑓 mean-flame
Slot Burners
• Slot burners are dependent on Froude number
• : Momentum-Controlled, Mixed (transitional), Buoyancy-Controlled
𝐽
• 𝐼 = 𝑒,𝑎𝑐𝑡 ; for plug nozzle velocity profile: 𝐼 = 1; for parabolic: 𝐼 = 1.5
𝑚𝐹 𝑣𝑒
• Need to iterate since we are trying to find 𝐿𝑓 (Initially guess = 1 m to find 𝐹𝑟𝑓 )
• Slot Nozzle are dependent on Froude number (stagnant oxidizer);
• 𝐹𝑟𝑓 =
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑗𝑒𝑡 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦
=
𝑣𝑒 𝐼𝑌𝐹,𝑠𝑡𝑖𝑜𝑐
𝑎𝐿𝑓
2
=
𝑣𝑒 𝐼𝑌𝐹,𝑠𝑡𝑖𝑜𝑐
0.6𝑔
1500𝐾
𝑇∞
2
−1 𝐿𝑓
• plug nozzle velocity profile: 𝐼 = 1; for parabolic: 𝐼 = 1.5; Initially guess = 1 m
4
• 𝐹𝑟𝑓 ≫ 1: Momentum Controlled: 𝐿𝑓,𝑒𝑥𝑝 = 8.6 ∙ 10
• 𝐹𝑟𝑓 ≪ 1: Buoyancy Controlled: 𝐿𝑓,𝑒𝑥𝑝 = 2 ∙
4
• 𝐹𝑟𝑓 ~ 1: Transitional: 𝐿𝑓 = 9 𝐿𝑓,𝑀
𝐿𝑓,𝐵
𝐿𝑓,𝑀
𝑏𝛽2 𝑄𝐹
𝑇∞ 2
~𝑄𝐹
𝐻𝐼𝑌𝑓,𝑆𝑡𝑜𝑖𝑐 𝑇𝐹
4 4 4
3 𝛽 𝑄𝐹 𝑇∞
10 𝑎ℎ4 𝑇 4
𝐹
3
1 + 3.38
1 3
𝐿𝑓,𝑀
𝐿𝑓,𝐵
= 2000
;𝛽 =
𝑄𝐹 𝛽𝑇∞
ℎ𝑇𝐹
1
4 𝑖𝑛𝑣𝑒𝑟𝑓
4
1 3
1
𝑎
1
1+𝑆
4
~𝑄𝐹 3
3 2 3
−1
• These are independent of 𝒟∞ mean diffusion coefficient for oxidizer at stream temperature 𝑇∞
Geometry and Flow Rate Dependence
• Page 341, Fig. 9.9,
• Methane
• Same areas
• 𝐿𝑓 increases with 𝑄𝐹
• Circular: 𝐿𝑓 ~𝑄𝐹
4
• Slot, 𝐹𝑟𝑓 ≪ 1: Buoyancy-Controlled: 𝐿𝑓 ~𝑄𝐹 3
• 𝐿𝑓 decreases for large aspect ratios
Burning Fuel Jet
(Diffusion Flame)
• Laminar Diffusion flame structure
• T and Y versus r at different x
• Flame shape
• Assume flame surface is located
where Φ ≈ 1, stoichiometric mixture
• No reaction inside or outside this
• Products form in the flame sheet
and then diffuse inward and
outward
• No oxidizer inside the flame envelop
• Over-ventilated: enough oxidizer to
burn all fuel
Fuel
𝜌𝑒 , 𝑣𝑒 , 𝜇
𝑄𝐹 = 𝑣𝑒 𝜋𝑅2
𝑚𝐹 = 𝜌𝑒 𝑣𝑒 𝜋𝑅2