Transcript Slides

ME 475/675 Introduction to
Combustion
Lecture 38
Factors affecting diffusion flame length
Announcements
• HW 16 Ch. 9 (8, 10,12)
• Due Wed, 12/2/2015
• Term Project (3% of grade)
• http://wolfweb.unr.edu/homepage/greiner/teaching/MECH.475.675.Combu
stion/TermProjectAssignment.pdf
• Course Evaluations:
• log in with your NetID to www.unr.edu/evaluate
• Please complete before 11:59 PM on Wed, Dec. 9, 2015 (dead day)
• I personally think you should wait until the last week of class, but it’s more
important to me and the Department that you do the evaluation.
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, 𝑌𝐹 =
𝑅
0.375𝑅𝑒𝑗
𝑥
3𝜌𝑒 𝐽𝑒 1 2 1 𝑟
Dimensionless Similarity Variable: 𝜉 =
16𝜋
𝜇𝑥
𝜌 𝑣 𝑅
𝑣 𝑅
𝑣 𝑅
Jet Reynold number: 𝑅𝑒𝑗 = 𝑒 𝑒 = 𝑒 = 𝑒
𝜇
𝜈
𝒟
1+
𝜉2
4
−2
• Simple flame length model
• x = 𝐿𝐹 where 𝑌𝐹 = 𝑌𝐹,𝑠𝑡 at 𝑟 = 𝜉 = 0
3
8
• 𝐿𝐹 = 𝑅𝑒𝑗
𝑅
𝑌𝐹,𝑠𝑡
=
3 𝜌𝑒 𝑣𝑒 𝑅
𝑅𝜋
8
𝜇
𝑌𝐹,𝑠𝑡 𝜋
=
3 𝜌𝑒 𝑄𝐹
8𝜋 𝜇𝑌𝐹,𝑠𝑡
=
3 𝑚𝐹
8𝜋 𝜇𝑌𝐹,𝑠𝑡
=
3 𝑄𝐹
8𝜋 𝜈𝑌𝐹,𝑠𝑡
• Increases with 𝑄𝐹 = 𝑣𝑒 𝜋𝑅2 (not dependent on 𝑣𝑒 𝑜𝑟 𝑅 separately)
• What is the effect of buoyancy?
=
3 𝑄𝐹
8𝜋 𝒟𝑌𝐹,𝑠𝑡
Buoyancy effects
• Buoyancy causes differences between a non-reacting fuel jet
and a burning flame
• Makes the flow accelerate and narrows its shape
𝑑𝑌𝐹
𝑑𝑟
𝑑𝑢𝑥
𝑎𝑛𝑑
𝑑𝑟
• The narrowed flame has higher radial gradients
more momentum and species diffusion than fuel jets.
and
• Slows the fuel jet and shortens the length where 𝑌𝐹 = 𝑌𝐹,𝑆𝑡
• Buoyancy and diffusion effects on flame length “tend” to cancel,
allowing models that neglect both to be somewhat correct (within
order of magnitude)
Experimentally-Confirmed Numerical Solutions
• Roper Correlations pp. 336-9; Table
9.3, Equations 9.59 to 9.70
• Subscripts:
• thy = Theoretical
• expt = Experimental (use these)
• Experimental results
• round nozzles
• 𝐿𝑓,𝑒𝑥𝑝 = 1330
𝑄𝐹 𝑇∞ 𝑇𝐹
ln(1+1 𝑆)
• square nozzles
• 𝐿𝑓,𝑒𝑥𝑝 = 1045
• Inverse Gaussian
error function
“inverf” from
Table 9.4
𝑄𝐹 𝑇∞ 𝑇𝐹
𝑖𝑛𝑣𝑒𝑟𝑓 1+𝑆 −0.5 2
• Metric units (𝐿𝑓 m, 𝑄𝐹 m3/s)
• S = Stoichiometric Air to Fuel Molar
ratio
• 𝑆 = 4.76 𝑥 +
𝑦
4
for CxHy fuel
• Temperatures: 𝑇∞ oxidizer, 𝑇𝐹 Fuel,
𝑇𝑓 mean-flame
Slot Burners
• Slot burners are dependent on Froude number (because of larger surface area per volume?)
• 𝐹𝑟𝑓 =
𝑣𝑒 𝐼𝑌𝐹,𝑠𝑡𝑖𝑜𝑐
𝑎𝐿𝑓
2
=
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑗𝑒𝑡 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦
• 𝐹𝑟𝑓 ≫ 1: Momentum-Controlled, 𝐹𝑟𝑓 ~ 1: Mixed (transitional), 𝐹𝑟𝑓 ≪ 1: Buoyancy-Controlled
• 𝑎 = 0.6𝑔
• 𝐼=
𝐽𝑒,𝑎𝑐𝑡
𝑚𝐹 𝑣𝑒
𝑇𝑓
𝑇∞
− 1 , 𝑇𝑓 = 1500𝐾
; for plug nozzle velocity profile: 𝐼 = 1; for parabolic: 𝐼 = 1.5
• Need to iterate since we are trying to find 𝐿𝑓 (Important hint: Initially guess = 1 m to find 𝐹𝑟𝑓 )
1
• Slot Nozzle Experimental results (stagnant oxidizer); 𝛽 =
4
• Momentum Controlled: 𝐿𝑓,𝑒𝑥𝑝 = 8.6 ∙ 10
• Buoyancy Controlled: 𝐿𝑓,𝑒𝑥𝑝 = 2 ∙ 103
4
• Transitional: 𝐿𝑓 = 9 𝐿𝑓,𝑀
𝐿𝑓,𝐵
𝐿𝑓,𝑀
1
4 𝑖𝑛𝑣𝑒𝑟𝑓 1+𝑆
𝑏𝛽2 𝑄𝐹
𝑇∞ 2
𝐻𝐼𝑌𝑓,𝑆𝑡𝑜𝑖𝑐 𝑇𝐹
4
𝛽4 𝑄𝐹4 𝑇∞
𝑎ℎ4 𝑇𝐹4
3
1 + 3.38
1 3
𝐿𝑓,𝑀
𝐿𝑓,𝐵
= 2000
𝑄𝐹 𝛽𝑇∞
ℎ𝑇𝐹
4
1 3
1
𝑎
(easier to use)
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 and square: 𝐿𝑓 ~𝑄𝐹
• Slot, 𝐹𝑟𝑓 ≫ 1: Momentum-Controlled: 𝐿𝑓 ~𝑄𝐹
4
• Slot, 𝐹𝑟𝑓 ≪ 1: Buoyancy-Controlled: 𝐿𝑓 ~𝑄𝐹 3
• 𝐿𝑓 decreases for large aspect ratios
• Buoyancy effects
• Square and circular ports aren’t affected by
buoyancy (momentum controlled, 𝐿𝑓 ~𝑄𝐹 )
• Slot can be affected by buoyancy (because
increases area increases drag?)
Example 9.3 page 339 (turn in next time for EC)
• It is desired to operate a square-port diffusion flame burner with a 50-mmhigh flame in a laboratory. Determine the volumetric flow rate required if
the fuel is propane. Also determine the heat release 𝑚Δℎ𝑐 of the flame.
What flow rate is required if methane is substituted for propane?
How will the flame length be affected if some other
oxidizer mixture is used instead of air: 𝜒𝑂2 ≠ 0.21
• Air is typically used as the Oxidizer mixture
• For air 𝜒𝑂2 = 0.21
• What do you expect if 𝜒𝑂2 increases
• Flame length will (a) increase, (b) decrease, or (c) stay the same?
• For a circular tube: 𝐿𝑓,𝑒𝑥𝑝 =
𝑄𝐹 𝑇∞ 𝑇𝐹
1330
ln(1+1 𝑆)
• What is affected by oxidizer 𝜒𝑂2 ?
• The stoichiometric air (oxidizer) to fuel molar ratio, 𝑆
Affect of oxidizer oxygen content on 𝐿𝑓 (𝜒𝑂2 < or > 21%)
• Circular tube
•
•
•
•
𝑄𝐹 𝑇∞ 𝑇𝐹
𝐿𝑓,𝑒𝑥𝑝 = 1330
ln(1+1 𝑆)
1
𝑦
𝑆=
𝑥+
𝜒𝑂2
4
2
For methane: 𝑆𝐶𝐻4 =
𝜒𝑂2
𝐿𝑓
𝐿𝑓,21%
=
ln(1+1 𝑆21% )
ln 1+1 𝑆
=
ln(1+0.21 2)
ln 1+𝜒𝑂2 2
• Increasing 𝜒𝑂2 in oxidizer decreases flame
length compared to air
• Circular tube
Fuel Dependence
• 𝐿𝑓,𝑒𝑥𝑝 = 1330
𝐶3 𝐻8
• 𝑆=
𝑁𝑂𝑥
𝑁𝐹𝑢 𝑆𝑡
𝑦
4
𝑂2 + 3.76𝑁2 → ⋯
𝑁 𝑂2
1
=𝜒
𝑁𝐶𝑥 𝐻𝑦
𝑂2
1
𝑆𝑡
=𝜒
𝑂2
𝑦
𝑥+4
• Alkane Fuels: 𝐶𝐻4 , 𝐶2 𝐻6 , 𝐶3 𝐻8 ,… 𝐶𝑥 𝐻2(𝑥+1)
2(𝑥+1)
4
= 4.76 1.5𝑥 + 0.5
• 𝑆𝐶𝐻4 = 9.52
• For a given flow rate 𝑄𝐹 and air oxidizer
𝐶𝐻4
•
𝐻2 𝑜𝑟 𝐶𝑂
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
• 𝑆 = 4.76 𝑥 +
𝐶2 𝐻6
(increases with S)
• For stoichiometric reaction of a generic HC fuel
• 𝐶𝑥 𝐻𝑦 + 𝑥 +
𝐶4 𝐻10
𝑄𝐹 𝑇∞ 𝑇𝐹
ln(1+1 𝑆)
𝐿𝑓
𝐿𝑓,𝐶𝐻4
=
ln(1+1 𝑆𝐶𝐻4 )
ln(1+1 𝑆)
,
• Heavier fuels require more air, and so more time and
distance (𝐿𝑓 ) to reach the stoichiometric condition
• Light fuels, 𝑆 =
4.76
2
= 2.33 (short flame)
1
• 𝐻2 + 2 𝑂2 + 3.76𝑁2 → 𝐻2 𝑂 + 1.88𝑁2
1
• 𝐶𝑂 + 2 𝑂2 + 3.76𝑁2 → 𝐶𝑂2 + 1.88𝑁2
Buoyancy effects
• Buoyancy causes differences between a non-reacting fuel jet and a burning flame
• Makes the flow accelerate and narrows its shape
• The narrowed flame has higher
𝑑𝑌𝐹
𝑑𝑟
and more diffusion than fuel jets.
• Buoyancy and diffusion effects on flame length “tend” to cancel, allowing models that
neglect both to be roughly correct (within order of magnitude)
• Variable-Density (and viscosity) Approximation (J. Fay)
• Assumes 𝜇 = 𝜇𝑟𝑒𝑓
• 𝐿𝐹,𝐵𝑢𝑜𝑦 ≈
•
•
•
•
𝑇
𝑇𝑟𝑒𝑓
3
𝑚𝐹
8𝜋 𝜇𝑟𝑒𝑓 𝑌𝐹,𝑠𝑡
𝜌∞
1
𝜌𝑟𝑒𝑓 𝐼 𝜌∞ 𝜌𝑓
𝜌∞ ambient density far from flame
𝜌𝑓 Flame density
Function 𝐼 𝜌∞ 𝜌𝑓 in table on page 335
𝜌𝑟𝑒𝑓 =? (turns out we can perform calculation without knowing it)
Buoyant Length Estimate
• 𝐿𝐹,𝐵𝑢𝑜𝑦 ≈
• 𝐿𝐹,𝐵𝑢𝑜𝑦 =
3
𝑚𝐹
𝜌∞
1
8𝜋 𝜇𝑟𝑒𝑓 𝑌𝐹,𝑠𝑡 𝜌𝑟𝑒𝑓 𝐼 𝜌∞ 𝜌𝑓
𝜌𝐹 𝜌∞
1
𝐿
2
𝜌𝑟𝑒𝑓
𝐼 𝜌∞ 𝜌𝑓 𝐹,𝐹𝑢𝑒𝑙𝐽𝑒𝑡
=
3
𝑄𝐹 𝜌𝐹
𝜌∞
1
8𝜋 𝜈𝑟𝑒𝑓 𝜌𝑟𝑒𝑓 𝑌𝐹,𝑠𝑡 𝜌𝑟𝑒𝑓 𝐼 𝜌∞ 𝜌𝑓
𝜌𝐹 = 𝐹𝑢𝑒𝑙 𝑑𝑒𝑛𝑠𝑖𝑡𝑦
• For hydrocarbon fueled flames:
• 𝜌𝐹 ≈ 𝜌∞
• 𝜌∞ 𝜌𝑓 ≈ 5,
• Table 9.2 gives 𝜌∞ 𝜌𝑟𝑒𝑓 = 3; 𝐼 𝜌∞ 𝜌𝑓 = 3.7
•
𝜌𝐹 𝜌∞
1
2
𝜌𝑟𝑒𝑓
𝐼 𝜌∞ 𝜌𝑓
=
𝜌∞ 𝜌∞
1
2
𝜌𝑟𝑒𝑓
𝐼 𝜌∞ 𝜌𝑓
=
32
3.7
= 2.4
• This buoyant model predicts flame length is 2.4 times longer than unburned
models
• But same order of magnitude
Momentum versus Buoyancy controlled flames
• Ratio of initial momentum to buoyancy can affect flame behavior
• Froude number
• 𝐹𝑟𝑓 =
𝑣𝑒 𝐼𝑌𝐹,𝑠𝑡𝑖𝑜𝑐
𝑎𝐿𝑓
• 𝑎 = 0.6𝑔
• 𝐼=
𝐽𝑒,𝑎𝑐𝑡
𝑚𝐹 𝑣𝑒
𝑇𝑓
𝑇∞
2
=
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑗𝑒𝑡 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦
− 1 , 𝑇𝑓 = 1500𝐾
; for plug nozzle velocity profile: 𝐼 = 1; for parabolic: 𝐼 = 1.5
• Need to iterate since we are trying to find 𝐿𝑓 (Initially guess = 1 m to find 𝐹𝑟𝑓 )
• Affect flames from slots, but not from square or round ducts.
• 𝐹𝑟𝑓 ≫ 1: Momentum-Controlled
• 𝐹𝑟𝑓 ≫ 1: Mixed (transitional)
• 𝐹𝑟𝑓 ≪ 1: Buoyancy-Controlled
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
• Roughly how long will the flame
be?
Fuel
𝜌𝑒 , 𝑣𝑒 , 𝜇
𝑄𝐹 = 𝑣𝑒 𝜋𝑅2
𝑚𝐹 = 𝜌𝑒 𝑣𝑒 𝜋𝑅2