1. ME 475/675 Introduction to Combustion
Lecture 39

2. Announcements HW16 Ch. 9 (8, 10,12) Term Project (3% of grade)
Due Wednesday, 12/2/2015
Term Project (3% of grade)
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3. Flame length (a measurable quantity)
For un-reacting fuel jet (no buoyancy)
𝐿 𝐹 = 3 8 𝑅𝑒 𝑗 𝑅 𝑌 𝐹,𝑠𝑡 = 3 8𝜋 𝑄 𝐹 𝜈 𝑌 𝐹,𝑠𝑡 = 3 8𝜋 𝑄 𝐹 𝒟 𝑌 𝐹,𝑠𝑡
Jet Reynold number: 𝑅𝑒 𝑗 = 𝜌 𝑒 𝑣 𝑒 𝑅 𝜇 = 𝑣 𝑒 𝑅 𝜈 = 𝑣 𝑒 𝑅 𝒟
Experimental Roper Correlations include buoayancy pp
round nozzles: 𝐿 𝑓,𝑒𝑥𝑝 =1330 𝑄 𝐹 𝑇 ∞ 𝑇 𝐹 ln⁡(1+ 1 𝑆) ~ 𝑄 𝐹
square nozzles: 𝐿 𝑓,𝑒𝑥𝑝 =1045 𝑄 𝐹 𝑇 ∞ 𝑇 𝐹 𝑖𝑛𝑣𝑒𝑟𝑓 1+𝑆 − ~ 𝑄 𝐹
Slot Nozzle in stagnant oxidizer are dependent on Froude number
𝐹𝑟 𝑓 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑗𝑒𝑡 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦 = 𝑣 𝑒 𝐼𝑌 𝐹, 𝑠𝑡𝑖𝑜𝑐 2 𝑎 𝐿 𝑓 = 𝑣 𝑒 𝐼𝑌 𝐹, 𝑠𝑡𝑖𝑜𝑐 𝑔 1500𝐾 𝑇 ∞ −1 𝐿 𝑓
Plug nozzle velocity profile: 𝐼=1; Parabolic profile : 𝐼=1.5; Initially guess = 1 m
𝐹𝑟 𝑓 ≫1: Momentum Controlled: 𝐿 𝑓,𝑒𝑥𝑝 =86,000 𝑏 𝛽 2 𝑄 𝐹 𝐻𝐼 𝑌 𝑓,𝑆𝑡𝑜𝑖𝑐 𝑇 ∞ 𝑇 𝐹 2 ~ 𝑄 𝐹 ; 𝛽= 1 4 𝑖𝑛𝑣𝑒𝑟𝑓 1 1+𝑆
𝐹𝑟 𝑓 ≪1: Buoyancy Controlled: 𝐿 𝑓,𝑒𝑥𝑝 = 𝑄 𝐹 𝛽 𝑇 ∞ ℎ 𝑇 𝐹 𝑎 ~ 𝑄 𝐹 4 3
𝐹𝑟 𝑓 ~ 1: Transitional: 𝐿 𝑓 = 4 9 𝐿 𝑓,𝑀 𝐿 𝑓,𝐵 𝐿 𝑓,𝑀 𝐿 𝑓,𝑀 𝐿 𝑓,𝐵 −1

4. Effect of Oxidizer Oxygen Content ( 𝜒 𝑂 2 < or > 21%)
Circular tube
𝐿 𝑓,𝑒𝑥𝑝 =1330 𝑄 𝐹 𝑇 ∞ 𝑇 𝐹 ln⁡(1+ 1 𝑆) =1330 𝑄 𝐹 𝑇 ∞ 𝑇 𝐹 ln 1+ 𝜒 𝑂 2 2
Stoichiometric moles of oxidizer per mole of fuel
𝑆= 1 𝜒 𝑂 2 𝑥+ 𝑦 4
For methane 𝑆 𝐶 𝐻 4 = 2 𝜒 𝑂 2
𝐿 𝑓, 𝜒 𝑂 𝐿 𝑓, 𝜒 𝑂 2 =21% = 𝑄 𝐹 𝑇 ∞ 𝑇 𝐹 ln 1+ 𝜒 𝑂 𝑄 𝐹 𝑇 ∞ 𝑇 𝐹 ln ⁡ ⁡ = ln⁡( ) ln 1+ 𝜒 𝑂 ⁡
Increasing 𝜒 𝑂 2 in oxidizer decreases flame length

5. Fuel Dependence Circular tube flame length
𝐿 𝑓,𝑒𝑥𝑝 =1330 𝑄 𝐹 𝑇 ∞ 𝑇 𝐹 ln⁡(1+ 1 𝑆) (increases with S)
For stoichiometric reaction of a generic HC fuel
𝐶 𝑥 𝐻 𝑦 + 𝑥+ 𝑦 4 𝑂 𝑁 2 →…
𝑆= 𝑁 𝑂𝑥 𝑁 𝐹𝑢 𝑆𝑡 = 1 𝜒 𝑂 𝑁 𝑂 2 𝑁 𝐶 𝑥 𝐻 𝑦 𝑆𝑡 = 1 𝜒 𝑂 2 𝑥+ 𝑦 4
If air is the oxidizer, then 𝜒 𝑂 2 =0.21 and 𝑆=4.76 𝑥+ 𝑦 4
If the oxidizer is pure 𝑂 2 , then 𝜒 𝑂 2 =1 and 𝑆=𝑥+ 𝑦 4
Alkane Fuels: 𝐶 𝐻 4 , 𝐶 2 𝐻 6 , 𝐶 3 𝐻 8 ,… 𝐶 𝑥 𝐻 2(𝑥+1) , y=2(𝑥+1)
𝑆=4.76 𝑥+ 2(𝑥+1) 4 = 𝑥+0.5
𝑆 𝐶 𝐻 4 =9.52
For a given flow rate 𝑄 𝐹 and air oxidizer
𝐿 𝑓 𝐿 𝑓,𝐶 𝐻 4 = ln⁡(1+ 1 𝑆 𝐶 𝐻 4 ) ln⁡(1+ 1 𝑆) = ln⁡(1.11) ln⁡ 𝑥 ,
Heavier fuels require more air, and so more time and distance ( 𝐿 𝑓 ) to reach the stoichiometric condition
Light fuels, 𝑆= =2.33 (short flame)
𝐻 𝑂 𝑁 2 → 𝐻 2 𝑂+1.88 𝑁 2
𝐶𝑂 𝑂 𝑁 2 → 𝐶𝑂 𝑁 2
𝐶 4 𝐻 10
𝐶 3 𝐻 8
𝐶 2 𝐻 6
𝐶 𝐻 4
𝐻 2 𝑜𝑟 𝐶𝑂

6. Stoichiometric Factors
When nozzle gas is pure 𝐶 𝑥 𝐻 𝑦 fuel and ambient gas is air with 𝑂 2 mole fraction 𝜒 𝑂 2
𝑆= 1 𝜒 𝑂 𝑁 𝑂 2 𝑁 𝐶 𝑥 𝐻 𝑦 𝑆𝑡 = 1 𝜒 𝑂 2 𝑥+ 𝑦 4 = 𝑆 𝑃𝑢𝑟𝑒 (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= 𝑁 𝐴𝑚𝑏𝑖𝑒𝑛𝑡 𝑁 𝑁𝑜𝑧𝑧𝑒𝑙 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 𝑁 2 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

7. Primary Aeration (of nozzle)
𝑆= 𝑁 𝐴𝑚𝑏𝑖𝑒𝑛𝑡 𝑁 𝑁𝑜𝑧𝑧𝑒𝑙 𝑆𝑡 = 1− 𝜓 𝑃𝑟𝑖 𝑆 𝑃𝑢𝑟𝑒 1+ 𝜓 𝑃𝑟𝑖 𝑆 𝑃𝑢𝑟𝑒 𝑆 𝑃𝑢𝑟𝑒 𝑆 𝑃𝑢𝑟𝑒
= 1− 𝜓 𝑃𝑟𝑖 1 𝑆 𝑃𝑢𝑟𝑒 + 𝜓 𝑃𝑟𝑖
𝐿 𝑓 𝐿 𝑓,𝑝𝑢𝑟𝑒 = ln⁡(1+ 1 𝑆 𝑃𝑢𝑟𝑒 ) ln 1+ 1 𝑆 ⁡ = ln⁡(1+ 1 𝑆 𝑃𝑢𝑟𝑒 ) ln 𝑆 𝑃𝑢𝑟𝑒 + 𝜓 𝑃𝑟𝑖 1− 𝜓 𝑃𝑟𝑖 ⁡
For methane 𝑆 𝑃𝑢𝑟𝑒 =9.52
Ambient
𝑁 𝐴𝑚𝑏 = 1− 𝜓 𝑃𝑟𝑖 𝑆 𝑃𝑢𝑟𝑒
Fuel+Oxidizer
𝐶 𝑥 𝐻 𝑦 + 𝜓 𝑃𝑟𝑖 𝑥+ 𝑦 4 𝑂 𝑁 2
𝑁 𝑁𝑜𝑧 =1+ 𝜓 𝑃𝑟𝑖 𝜒 𝑂 2 𝑥+ 𝑦 4 =1+ 𝜓 𝑃𝑟𝑖 𝑆 𝑃𝑢𝑟𝑒

8. Fuel Dilution
𝑆= 𝑁 𝐴𝑚𝑏𝑖𝑒𝑛𝑡 𝑁 𝑁𝑜𝑧𝑧𝑒𝑙 𝑆𝑡 = 1 𝜒 𝑂 2 𝑥+ 𝑦 𝑁 𝐷 = 𝑥+ 𝑦 4 𝜒 𝑂 𝑁 𝐷
𝜒 𝐷 = 𝑁 𝐷 1+ 𝑁 𝐷 = 𝑁 𝐷 +1 ; 1 𝜒 𝐷 = 1 𝑁 𝐷 +1; 𝑁 𝐷 = 𝜒 𝐷 −1
𝑁 𝐷 = 𝜒 𝐷 1− 𝜒 𝐷
𝑆= 𝑥+ 𝑦 4 𝜒 𝑂 𝜒 𝐷 1− 𝜒 𝐷 = 𝑥+ 𝑦 4 𝜒 𝑂 − 𝜒 𝐷 = 𝑥+ 𝑦 4 1− 𝜒 𝐷 𝜒 𝑂 2
𝐿 𝑓 𝐿 𝑓,𝑝𝑢𝑟𝑒 = ln⁡(1+ 1 𝑆 𝑃𝑢𝑟𝑒 ) ln 1+ 1 𝑆 ⁡ = ln⁡(1+ 1 𝑆 𝑃𝑢𝑟𝑒 ) ln 1+ 𝜒 𝑂 2 𝑥+ 𝑦 4 1− 𝜒 𝐷 ⁡
For methane 𝑆 𝑃𝑢𝑟𝑒 =9.52, 𝑥+ 𝑦 4 =2
For air 𝜒 𝑂 2 =0.21
𝐿 𝑓 𝐿 𝑓,𝑝𝑢𝑟𝑒 = ln⁡(1.105) ln − 𝜒 𝐷 ⁡
Ambient
𝑁 𝐴𝑚𝑏 = 𝑆 𝑃𝑢𝑟𝑒
= 𝑥+ 𝑦 4 𝜒 𝑂 2
Fuel+Oxidizer
𝐶 𝑥 𝐻 𝑦 + 𝑁 𝐷 𝐷

9. 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 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.

10. 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)
Applications: Spray Combustion (not droplet burning)
Oil home heaters (

11. Applications: Diesel Engines
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
Droplets evaporate and premix with air, burn then auto-ignite the rest of the mixture
Blows into main chamber and completes combustion
Injector
Pre-mix
chamber
Glow Plug

12. Gas Turbine Engines (aircraft and stationary)
Annular Combustor is a relatively small component

13. Annular Multistage Combustor
Fuel is atomize
Premixed and staged to avoid NOx formation
Walls are protected from high temperatures by film cooling

14. Liquid Rocket Engines (fuel and oxidizer are liquid)
Pressure-fed by high pressure gas
Pump-fed by turbo- pumps
Mixed by colliding jets to form unstable sheets and break up

16. Flame length (a measurable quantity)
Φ 𝑟=0,𝑥= 𝐿 𝑓 =1; 𝑌 𝐹 = 𝑌 𝐹,𝑠𝑡
For un-reacting fuel jet (no buoyancy)
For Schmidt number 𝑆𝑐= 𝜈 𝒟 =1, 𝑌 𝐹 =0.375 𝑅𝑒 𝑗 𝑅 𝑥 𝜉 −2
Dimensionless Similarity Variable: 𝜉= 3 𝜌 𝑒 𝐽 𝑒 16𝜋 𝜇 𝑟 𝑥
Jet Reynold number: 𝑅𝑒 𝑗 = 𝜌 𝑒 𝑣 𝑒 𝑅 𝜇 = 𝑣 𝑒 𝑅 𝜈 = 𝑣 𝑒 𝑅 𝒟
Flame length, x= 𝐿 𝐹 where 𝑌 𝐹 = 𝑌 𝐹,𝑠𝑡 at 𝑟=𝜉=0
𝑌 𝐹,𝑠𝑡 =0.375 𝑅𝑒 𝑗 𝑅 𝐿 𝐹 −2
𝐿 𝐹 = 3 8 𝑅𝑒 𝑗 𝑅 𝑌 𝐹,𝑠𝑡 = 𝜌 𝑒 𝑣 𝑒 𝑅 𝜇 𝑅𝜋 𝑌 𝐹,𝑠𝑡 𝜋 = 3 8𝜋 𝜌 𝑒 𝑄 𝐹 𝜇 𝑌 𝐹,𝑠𝑡 = 3 8𝜋 𝑚 𝐹 𝜇 𝑌 𝐹,𝑠𝑡 = 3 8𝜋 𝑄 𝐹 𝜈 𝑌 𝐹,𝑠𝑡 = 3 8𝜋 𝑄 𝐹 𝒟 𝑌 𝐹,𝑠𝑡
Increases with 𝑄 𝐹 = 𝑣 𝑒 𝜋 𝑅 2 (not dependent on 𝑣 𝑒 𝑜𝑟 𝑅 separately)
Decreases with increasing 𝒟 and 𝑌 𝐹,𝑠𝑡 = 𝑚 𝑂𝑥 𝑚 𝐹𝑢 = 𝑁 𝑂𝑥 𝑁 𝐹𝑢 𝑀𝑊 𝑂𝑥 𝑀𝑊 𝐹𝑢 = 1 1+𝑆 𝑀𝑊 𝑂𝑥 𝑀𝑊 𝐹𝑢 ;
Depend on fuel
For 𝐶 𝑥 𝐻 𝑦 fuel, 𝑆=4.76 𝑥+ 𝑦 4 , 𝑀𝑊 𝑂𝑥 𝑀𝑊 𝐹𝑢 = 𝑥 𝑦
For y = 2x+2 (alkanes), decreases with increasing x
What about 𝒟?
What is the effect of buoyancy?

17. Experimentally-Confirmed Numerical Solutions
Roper Correlations pp ; Table 9.3, Equations 9.59 to 9.70
Subscripts:
thy = Theoretical
expt = Experimental
Experimental results
round nozzles,
𝐿 𝑓,𝑒𝑥𝑝 =1330 𝑄 𝐹 𝑇 ∞ 𝑇 𝐹 ln⁡(1+ 1 𝑆) ~ 𝑄 𝐹
square nozzles,
𝐿 𝑓,𝑒𝑥𝑝 =1045 𝑄 𝐹 𝑇 ∞ 𝑇 𝐹 𝑖𝑛𝑣𝑒𝑟𝑓 1+𝑆 − ~ 𝑄 𝐹
Metric units (m, m3/s)
S = Molar Stoichiometric ratio = 4.75*(x+y/4) for CxHy fuel
Temperatures: 𝑇 ∞ oxidizer, 𝑇 𝐹 Fuel, 𝑇 𝑓 mean-flame
Inverse Gaussian error function “inverf” from Table 9.4

18. Slot Burners Slot burners are dependent on Froude number
: Momentum-Controlled, Mixed (transitional), Buoyancy-Controlled
𝐼= 𝐽 𝑒,𝑎𝑐𝑡 𝑚 𝐹 𝑣 𝑒 ; for plug nozzle velocity profile: 𝐼=1; for parabolic: 𝐼=1.5
Need to iterate since we are trying to find 𝐿 𝑓 (Initially guess = 1 m to find 𝐹𝑟 𝑓 )
Slot Nozzle are dependent on Froude number (stagnant oxidizer);
𝐹𝑟 𝑓 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑗𝑒𝑡 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦 = 𝑣 𝑒 𝐼𝑌 𝐹, 𝑠𝑡𝑖𝑜𝑐 2 𝑎 𝐿 𝑓 = 𝑣 𝑒 𝐼𝑌 𝐹, 𝑠𝑡𝑖𝑜𝑐 𝑔 1500𝐾 𝑇 ∞ −1 𝐿 𝑓
plug nozzle velocity profile: 𝐼=1; for parabolic: 𝐼=1.5; Initially guess = 1 m
𝐹𝑟 𝑓 ≫1: Momentum Controlled: 𝐿 𝑓,𝑒𝑥𝑝 =8.6∙ 𝑏 𝛽 2 𝑄 𝐹 𝐻𝐼 𝑌 𝑓,𝑆𝑡𝑜𝑖𝑐 𝑇 ∞ 𝑇 𝐹 2 ~ 𝑄 𝐹 ; 𝛽= 1 4 𝑖𝑛𝑣𝑒𝑟𝑓 1 1+𝑆
𝐹𝑟 𝑓 ≪1: Buoyancy Controlled: 𝐿 𝑓,𝑒𝑥𝑝 =2∙ 𝛽 4 𝑄 𝐹 4 𝑇 ∞ 4 𝑎 ℎ 4 𝑇 𝐹 = 𝑄 𝐹 𝛽 𝑇 ∞ ℎ 𝑇 𝐹 𝑎 ~ 𝑄 𝐹 4 3
𝐹𝑟 𝑓 ~ 1: Transitional: 𝐿 𝑓 = 4 9 𝐿 𝑓,𝑀 𝐿 𝑓,𝐵 𝐿 𝑓,𝑀 𝐿 𝑓,𝑀 𝐿 𝑓,𝐵 −1
These are independent of 𝒟 ∞ mean diffusion coefficient for oxidizer at stream temperature 𝑇 ∞

19. Geometry and Flow Rate Dependence
Page 341, Fig. 9.9,
Methane
Same areas
𝐿 𝑓 increases with 𝑄 𝐹
Circular: 𝐿 𝑓 ~ 𝑄 𝐹
Slot, 𝐹𝑟 𝑓 ≪1: Buoyancy-Controlled: 𝐿 𝑓 ~ 𝑄 𝐹 4 3
𝐿 𝑓 decreases for large aspect ratios

20. 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
𝜌 𝑒 , 𝑣 𝑒 ,𝜇
𝑄 𝐹 = 𝑣 𝑒 𝜋 𝑅 2
𝑚 𝐹 = 𝜌 𝑒 𝑣 𝑒 𝜋 𝑅 2