Report

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 Heat and Radical • 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: ___?