Thermal theory of flame propagation

kinetic foundation of thermal flame theory and difference between combustion and flame
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Published Date:23-07-2017
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Laminar Premixed Flames: Flame Structure CEFRC Combustion Summer School 2014 Prof. Dr.-Ing. Heinz Pitsch Copyright ©2014 by Heinz Pitsch. This material is not to be sold, reproduced or distributed without prior written permission of the owner, Heinz Pitsch. Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Thermal flame theory • Fundamentals and mass • Flame thickness and flame time balances of combustion systems • Four-step model for methane-air flames • Thermodynamics, flame temperature, and equilibrium • Three-step model for methane-air flames • Governing equations • Asymptotic structure of • Laminar premixed flames: stoichiometric methane-air flames Kinematics and burning velocity • Analytic expression for the burning • Laminar premixed flames: velocity of methane and other Flame structure hydrocarbon flames • Laminar diffusion flames • Flammability limits 2 Thermal Flame Theory The first theoretical treatment of stationary one dimensional flames: Thermal Flame Theory of Zeldovich and Frank-Kamenetzki (1938) • A classical example of a mathematical description of the combustion of premixed gases • Assumption of a one step global reaction with high activation energy → Mathematically solveable form of the governing equations • This theory is the origin of a development, which placed combustion science on a mathematical basis 3 Thermal Flame Theory • Starting point:  Stationary and flat flame front  Absolute values of the flow velocity and the burning velocity are identical, and in opposite direction Temperature and concentration profiles, schematically for a lean mixture Preheat zone Y F,u Reaction zone F 4 Thermal Flame Theory Flame front (lean mixture): • Complete fuel conversion • Excess oxygen remains • Temperature rises from the initial value T to the u Preheat zone adiabatic flame temperature T b • The area around the flame front is divided in Y F,u three zones by Zeldovich Reaction zone F and Frank-Kamenetzki:  Preheat zone  Reaction zone  Equilibrium zone 5 Thermal Flame Theory • Heat conduction out of reaction zone → Mixture in the preheat zone is continuously heated • Diffusion of combustion products Preheat zone back into the unburnt mixture • Diffusion of reactants (fuel and Y F,u oxygen), into the reaction zone F Reaction zone • Transition  Position x i  Temperature T i • Reaction zone: chemical reactions 6 Thermal Flame Theory • Simplification:  Global reaction equation  Reaction rate • Behind the flame front:  No chemical conversion → Reaction rate = 0  Complete consumption of: • Fuel (for lean mixtures) • Oxygen (for rich mixtures) • Both reactants (for stoichiometric mixtures)  In the equilibrium zone: 7 Thermal Flame Theory • Further simplifications:  Specific heat capacity is constant and identical for all components  Constant enthalpy of reaction  Lewis-Number Le = 1 → Enthalpy is constant • Coupling functions between the concentrations and the temperatures: 8 Thermal Flame Theory • Momentum equation, limit of small Mach-Numbers → constant pressure • With the ideal gas law and the coupling functions:  Density  Thermal conductivity  Reaction rate • as function of temperature • Solution of the continuity equation for one dimensional flows: 9 Thermal Flame Theory • Only one differential equation: Temperature equation in x-direction • Solution of this equation with the following assumptions suggested by Zeldovich and Frank-Kamenetzki:  Preheat zone, T T : no reactions: w = 0 i  Reaction zone, T T : Convective term on the left hand side is negligible compared to i the diffusive term and the reaction term • Validity of the second assumption:  Asymptotic theory  Nature of the reaction zone: very thin boundary layer  Mathematical justification by singular asymptotic expansion 10 Thermal Flame Theory • First assumption yields • Integration of the simplified differential equation in the preheat zone • For the first derivative, with boundary conditions at At 11 Thermal Flame Theory • Second assumption yields • Heat conduction term can be substituted by • The differential equation then becomes 12 Thermal Flame Theory • Integration of the differential equation with BC: At • Zeldovich und Frank-Kamenetzki: derivatives of the temperature in the preheat zone and reaction zone at position x are equal: i • Relations for preheat and reaction zone have boundary conditions at infinity and need to be equal  Eigenvalue  Burning velocity 13 Thermal Flame Theory • Evaluation of the integral in complete form is possible, only if further simplifications are introduced • Series expansion of the exponential term in centered at T and neglecting terms of higher order leads to: b 14 Thermal Flame Theory • Reaction zone: T differs only slightly from T b • Introduce dimensionless temperature  Even for high , order of magnitude is one • In the reaction zone: material properties assumed constant • Reaction becomes 15 Thermal Flame Theory • Integration yields: 16 Thermal Flame Theory • Integral: • With: Preheat zone  Asymptotic limit of high activation energies Y  Overlapping process of the solution F,u Reaction zone from preheat and reaction zones F • Substitution of q by q i u → Assumption: Reaction zone also valid in the preheat zone 17 Thermal Flame Theory • Physical explanation: Below T , the integral of the reaction rate is negligible i because of the high temperature dependence → No difference between integration from T to T or from T to T i b u b • q has a high negative values for high activation energies u • In terms, which include exp(q ), q can be replaced by u u → Terms disappear 0 0 At 18 Thermal Flame Theory Preheat zone: • Replacing T by T and l by l i b i b P  Preheat zone • Assumptions:  Reaction zone is so thin that the Y F,u preheat zone reaches till T b Reaction zone  T differs only minimally from T F i b 19 Thermal Flame Theory • Equating the following expressions leads to: with • Terms in S depend on equivalence ratio f:  Lean mixture  Oxygen mass fraction high, Y ≈ 0 B,b  Rich mixture  Fuel mass fraction high, Y ≈ 0 O ,b 2  Stoichiometric mixture  Y ≈ Y ≈ 0 O ,b B,b 2 20

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