AME 514 Applications of Combustion

1. AME 514Applications of Combustion Lecture 3: Combustion Theory II

2. Outline • Conservation equations revisited • 1D laminar premixed flame • Assumptions • Analysis • Results AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

3. Conservation of energy and species • Recall from AME 513, lecture 7 (Conservation equations) • Recall Le = k/rCPD is dimensionless– generally for gases D ≈ k/rCP ≈n, where k/rCP = a = thermal diffusivity, n = kinematic viscosity (“viscous diffusivity”) AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

4. 1D laminar premixed flame - formulation • Also recall that while is constant, i.e. doesn’t vary with reaction • If Le is not exactly 1, small deviations in Le (thus T) will have large impact on w due to high activation energy • Energy equation may have heat loss in q’’’ term, not present in species conservation equation • Special case: 1D, steady (∂/∂t = 0), constant CP (thus ∂h/∂T = CP∂T/∂t) & constant k, rD; simplified form of Arrhenuis expression (single reactant, far from stoichiometric) • Note that Z is not the usual one based on molar concentrations,but rather based on fuel mass fraction (units 1/time) AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

5. 1D laminar premixed flame - formulation AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

6. 1D laminar premixed flame - solution • How to solve? • Boundary conditions: x = -∞, T = 0; x = +∞, T = 1 • Cold boundary problem – reactants occurs even at x = -∞, so are already completely reacted by x = 0, so need to assume finite domain with non-zero dT/dx slope at inflow end (equivalent to assuming a small heat loss at cold boundary) • Can’t assume dT/dx = 0 at cold boundary, reaction is too slow and would take enormous domain to reach flame front • Need to see how different values of dT/dx at cold boundary affect solution AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

7. 1D premixed flame – numerical method • x = 0 is cold boundary (T = 0), assume finite dT/dx • “Guess” eigenvalue L • Fixed grid spacing Dx, at every subsequent grid point • Does T  1 as x  +∞? If not, adjust guess for L AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

8. 1D premixed flame – results For typical b = 10, e = 0.2, L= 0.2 AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

9. 1D premixed flame – results • What if Le ≠ 1? T ≠ 1 – Y, energy and species equations are not identical and are coupled AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

10. 1D premixed flame – structure • Outside of a thin reaction zone at x = 0 • Within reaction zone – temperature does not increase despite heat release – temperature acts to change slope of temperature profile, not temperature itself AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

11. Schematic of deflagration (from Lecture 1) • Temperature increases in convection-diffusion zone or preheat zoneahead of reaction zone, even though no heat release occurs there, due to balance between convection & diffusion • Temperature constant downstream (if adiabatic) • Reactant concentration decreases in convection-diffusion zone, even though no chemical reaction occurs there, for the same reason AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

12. Conservation equations - comments • In limit of infinitely thin reaction zone, T does not change but dT/dx does; integrating across reaction zone • Note also that from temperature profile: • Thus, change in slope of temperature profile is a measure of the total amount of reaction – but only when the reaction zone is thin enough that convection term can be neglected compared to diffusion term AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

13. Deflagrations - burning velocity • More rigorous analysis (Bush & Fendell, 1970) using Matched Asymptotic Expansions • Convective-diffusive (CD) zone (no reaction) of thickness d • Reactive-diffusive (RD) zone (no convection) of thickness d/b(1-e) where 1/[b(1-e)] is a small parameter • T(x) = T0(x) + T1(x)/[b(1-e)] + T2(x)/[b(1-e)]2 + … • Collect terms of same order in small parameter • Match T & dT/dx at all orders of b(1-e) where CD & RD zones meet • Same form as simple estimate (SL ~ {w}1/2, where w ~ Ze-bis an overall reaction rate, units 1/s), with additional constants • Why b-2term on reaction rate? • Reaction doesn’t occur over whole flame thickness d, only in thin zone of thickness d/b • Reactant concentration isn’t at ambient value Yi,∞, it’s at 1/b of this since temperature is within 1/b of Tad AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II

14. Deflagrations - burning velocity • What if not a single reactant, or Le ≠ 1? Mitani (1980) extended Bush & Fendell for reaction of the form where A is the deficient reactant, e.g. fuel in a lean mixture, resulting in • Recall order of reaction (n) = A+B • Still same form as simple estimate, but now b-(n+1)term since n may be something other than 1 (as Bush & Fendell assumed) • Also have LeA-nA and LeB-nB terms – why? For fixed thermal diffusivity (a), for higher LeA, DA is smaller, gradient of YA must be larger to match with T profile, so concentration of A is higher in reaction zone AME 514 - Spring 2013 - Lecture 3 - Combustion Theory II