Interactive Combustion in a Linear Array of 2D Laminar Isolated and Triple Burner Jets

Table 1

Summary of single, laminar jet results for 2D planar jet under non-buoyant conditions.

Row#	Parameter	2D Jet	Remarks
1	Properties	Variable	For 2D, 𝜌^2𝐷 = constant and; 𝜌^2𝜈 = constant.
2	𝑀^∗	𝑀^∗=6/5 for a parabolic 𝜈𝑥,𝑖 profile	forces important. 𝑀^∗=𝑀/(𝜌𝑖𝑣𝑥,𝑖𝑑^2𝑖), where M is the axial momentum flux rate at the inlet
3	𝐽^∗𝑏	𝐽^∗𝑏=6/5 for a parabolic 𝜈𝑥,𝑖 and parabolic 𝑏 profile	𝐽^∗𝑏=𝐽𝑏/(𝜌𝑑𝑖𝑣𝑥,𝑖(𝑏𝑖−𝑏∞))
		𝐽^∗𝑏=1 for a parabolic 𝜈𝑥,𝑖 and flat 𝑏 profile
		𝐽^∗𝑏=1 for a flat 𝜈𝑥,𝑖 and flat 𝑏 profile
4	Stretched coordinate	𝑦^′ = ∫^𝑦0𝜌/𝜌𝑖𝑑𝑦	For an ideal gas with a constant molecular weight 𝜌/𝜌𝑖=𝑇𝑖/𝑇
5	Similarity Coordinate, η [11]	𝜂=𝑀^∗1/3(𝑦^∗0𝑥02215𝑥^∗2/3)(𝑅𝑒𝑖/3)^2/3
		𝑦^∗=𝑦^′/𝑑𝑖
6	Modified Similarity variable, ξ	𝜉=(1/2)𝑀^∗1/3(3/2)^1/3𝜂=((1/2)𝑀^∗(3/2)^1/3(𝑦^∗/𝑥^∗2/3)(𝑅𝑒𝑖/3)^2/3=0.2752𝑅𝑒𝑖^2/3𝑦^∗𝑀^∗1/3/𝑥^∗2/3
7	Momentum equation in η and other solutions	𝑓^′2+𝑓𝑓^′′+𝑓^′′′=0
		𝑓^′ = (1/2) (3/2)^(2/3) Sech^2ξ
		f = (3/2)^1/3tanh ξ
		𝑣^∗𝑥=𝑓^′(𝜉) 𝑀^∗2/3 {𝑅𝑒𝑖0𝑥02215(3𝑥^∗)}^1/3
		𝑣^∗𝑦= [{𝑀^∗0𝑥02215𝑅𝑒𝑖}^1/30𝑥02215{(3𝑥)^∗2/3}]{2𝑓^′(𝜉)−𝑓(𝜉)}
8	Species equation in ψ	𝜓^” + Sc 𝑓^′ 𝜓 + 𝑆𝑐𝑓𝜓^′ = 0; where 𝜓=𝜑𝑥^∗1/3
9	Axial velocity; 𝑣^∗𝑥 = 𝑣𝑥/𝑣𝑥,𝑖	𝑣^∗𝑥 = 0.4543 𝑀^∗2/3 {𝑅𝑒𝑖0𝑥02215𝑥^∗}^1/3 Sech^2𝜉 𝑣^∗𝑥,max = 0.4543 𝑀^∗2/3 (𝑅𝑒𝑖0𝑥02215𝑥^∗)^1/3
10	Lateral velocity 𝑣^∗𝑦	𝑣^∗𝑦 = 0.5503{(𝑀^∗0𝑥02215𝑅𝑒𝑖)^1/30𝑥02215𝑥^∗2/3}{2 𝜉 sech^2𝜉−tanh𝜉}
		𝑣^∗𝑦 = 0 at 𝜉 = 0 and 𝜉 = 1.0887
		𝑣^∗∞,max= 0.5503 {(𝑀^∗/𝑅𝑒𝑖)^1/3/𝑥^∗2/3}
11a	Mixing layer thickness, 𝑦^∗𝑚, 𝑣𝑥/𝑣𝑥,max = 0.01	𝑦^∗𝑚 = 10.8765 𝑥^∗2/3/{𝑅𝑒𝑖^2/3𝑀^∗1/3}	Note mix layer varies as 𝑥^2/3 for 2D jet
11b	Jet half-width,; 𝑦^∗1/2 where 𝑣𝑥/𝑣𝑥0 = 1/2	𝜉1/2 = 0.8814; 𝑦^∗1/2= 3.203 𝑥^∗2/3/{𝑅𝑒𝑖^2/3𝑀^∗1/3}, jet angle; 𝛼 = tan^−1{𝑦^∗1/2/𝑥^∗)
11c	Velocity contour 𝑣^∗𝑥=𝑆/𝑣𝑥,𝑖	𝑣^∗𝑦 = 3.6337{𝑥^∗2/3/(𝑅𝑒𝑖^2/3𝑀^∗1/3)}×Sech^−1
		{1.4836𝑥^∗1/6(𝑆/𝑣𝑥,𝑖)^1/2/(𝑅𝑒𝑖^1/6𝑀^∗1/3}
12	Mass flow within or 𝑚(𝑦,𝑥)	𝑚̇′(𝑦^∗,𝑥^∗)/𝑚̇^′𝑖={3.3016 𝑥^∗1/3 𝑀^∗1/3/𝑅𝑒𝑖^1/3} tanh 𝜉	Used for estimating the mass flow within 𝑃/2 for multiple burners
13	Total mass flow 𝑚(𝑥)	𝑚̇′(∞,𝑥^∗)/𝑚̇^′𝑖={3.3016𝑥^∗𝑀^∗/𝑅𝑒𝑖)^1/3}
14	Height at which two adjacent mixing layers intersect	𝑥^∗ = 0.009856 𝑃^∗3/2 {𝑅𝑒𝑖𝑀^∗1/2}	Burners are located 𝑙 apart
15a	Air entrained at 𝑥	𝑚̇^′𝑎/𝑚̇^′𝑖={3.3016 (𝑥^∗ 𝑀^∗/𝑅𝑒𝑖)^1/3}−1
15b	Air entrained; at 𝑥 = mixing layer intersection height	𝑚̇^′𝑎/𝑚̇^′𝑖={0.70789 (2𝑦^∗𝑚)^/2𝑀^∗1/6}−1,
		𝑚̇^′𝑎/𝑚̇^′𝑖={0.70789𝑃^∗/2𝑀^∗1/6}−1,	Buoyancy affects air entrained for 2D
		within interburner spacing/injected flow for which 𝑦^∗𝑚 = 𝑃^∗/2
15c	Air entrained at x = 𝐻stoich	𝑚̇^′𝑎/𝑚̇^′𝑖 =1.5𝐶𝐽^∗/(𝜙𝑓)−1 𝑚̇^′𝑎/𝑚̇^′𝑖 ≈1.5𝐶𝐽^∗/(𝜙𝑓)≈𝐶; 𝑚̇^′𝑎(𝑆𝑐)/𝑚̇^′𝐴(𝑆𝑐=1)≈𝐶
16	Species and non dimensional Shvab-Zeldovich,; 𝜑=(𝑏−𝑏∞)/(𝑏𝑖−𝑏∞)	𝜑(𝑆𝑐=1)={0.45428(𝑅𝑒𝑖/𝑥^∗)^1/3(𝐽^∗/𝑀^∗1/3)}×Sech^2𝜉 𝑁(𝑆𝑐±1)={0.45428𝐶(𝑅𝑒𝑖/𝑥^∗)^1/3(𝐽^∗/𝑀^∗1/3)}×Sech^2𝑆𝑐 𝜉; 𝐶=−0.0294𝑆𝑐^2+0.3534𝑆𝑐+2/3	𝑏=𝑌𝐹,𝑌𝑂2 for mixing, 𝑏={𝑌𝐹−𝑌𝑂2/𝜈𝑂2},{𝑌𝐹+𝐶𝑝(𝑇−𝑇𝑖)/ℎ𝑐},{𝑌𝑂2/𝜈𝑂2+𝐶𝑝(𝑇−𝑇𝑖)/ℎ𝑐} for species
17a	Height of Contours, 𝐻^∗	𝐻^∗st= 𝐻^∗𝑓 = 0.09375𝑅𝑒𝑖C ^3(𝐽^∗3/𝑀^∗)/𝜑^3𝑓,; 𝐻^∗𝑣 = 0.09375 𝑅𝑒𝑖 𝑀^∗2 𝑣^3𝑖/𝑆^3	𝜑𝑓=𝑠/(1+𝑠), 𝑠=𝑌𝑂2,∞/(Λ𝑂2𝑌𝐹,𝐼) 𝜑𝑓0.0544 for pure Methane in air; 𝐻^∗st = 𝐻^∗𝑓 for combustion 𝐻^∗=𝐻/𝑑𝑖
18	Maximum width of flame, 𝑦^∗𝑓,max	𝑦^∗𝑓,max ={0.7499(𝛾Sech^4𝑆𝑐𝛾)} (𝐽^∗2/𝑀^∗4/3)(C/𝜑𝑓)^2; (valid when 𝜌 is constant 0<𝑦<𝑦𝑓)
19	Ratio of flame height and max width (𝐻^∗/𝑦^∗𝑓,max)	𝐻^∗𝑓/𝑦^∗𝑓,max =0.125𝑅𝑒𝑖𝐶(𝐽^∗𝑀^∗1/3)/{𝜑𝑓 (Sech^4𝑆𝑐γ )}(valid when ρ is constant and 0<𝑦<𝑦𝑓)
20	Distance 𝑥^∗𝑓,mw at max width	𝑥^∗𝑓,mw = 𝑥^∗𝑓,mw Sech^6𝑆𝑐 𝛾, where,; 𝛾tanh𝛾=1/(4𝑆𝑐)transcendental equation (valid when ρ is constant and 0<𝑦<𝑦𝑓)
21	𝑥^∗𝑓,𝑚𝑤/𝐻^∗flame	𝑥^∗𝑓,mw /𝐻^∗𝑓 = Sech^6𝑆𝑐 (𝛾𝑆𝑐) (valid when ρ is constant and 0<𝑦<𝑦𝑓)
22	Flame angle with axis, 𝜃^′, at flame tip and location 𝑥^∗(Figure 4(b))	tan 𝜃^′tip =1/(1.6512 𝑆𝑐 𝐻𝑓^∗1/3 𝑅𝑒𝑖^2/3 𝑀^∗1/3); where,
		tan 𝜃^′tip =(tan𝜃^′)tip 𝜌𝑓/𝜌𝑡⋅{tan𝜃^′}tip(𝑇𝑖/𝑇𝑓)
		tan 𝜃^′tip = {𝐻^∗𝑓/𝑥^∗}^1/3−1/6𝑆𝑐 tan 𝜃^′tip+ (2/3)
		Cosh^−1(𝐻^∗𝑝𝑙/𝑥^∗)^1/6Sc (1/𝑥^∗1/3)/(0.2752𝑅𝑒𝑖^2/3 𝑀^∗1/3)
		and 𝜃^′tip = 𝜃 for constant 𝜌
23	Lift-off Height, {𝐿^∗ /𝐻^∗flame}	𝐿^∗/𝐻^∗𝑓 =[{S/𝑣,𝑖} {𝐶𝐽^∗/(𝜑𝑓𝑀^∗)}]^3𝑆𝑐/(1−𝑆𝑐)	Set 𝐿^∗ = 𝐻^∗𝑓 for blow-off; 𝐿^∗= lift-off height for 𝑆𝑐>1, 𝐿^∗=𝐿^∗ig ignitable height for 𝑆𝑐<1
		For 𝑆𝑐 = 1, 𝐿^∗→∞
24	Blow-off velocity	𝑣blow = 𝑆𝐶𝐽^∗/(𝜑𝑓𝑀^∗)