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Laminar Premixed Flames: Flame Structure

Laminar Premixed Flames: Flame Structure
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 • Fourstep model for methaneair flames • Thermodynamics, flame temperature, and equilibrium • Threestep model for methaneair flames • Governing equations • Asymptotic structure of • Laminar premixed flames: stoichiometric methaneair 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 FrankKamenetzki (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 FrankKamenetzki:  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  LewisNumber Le = 1 → Enthalpy is constant • Coupling functions between the concentrations and the temperatures: 8 Thermal Flame Theory • Momentum equation, limit of small MachNumbers → 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 xdirection • Solution of this equation with the following assumptions suggested by Zeldovich and FrankKamenetzki:  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 FrankKamenetzki: 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 Thermal Flame Theory • In a stoichiometric mixture, the last term is dominant  Approximation: 21 Thermal Flame Theory (Summary) • Preheat zone: reaction rate is neglected • Reaction zone: convection term is neglected • Approximation of the reaction rate by a series expansion centered at T , only b the exponential term is expanded • Material properties are set to their values at T b • Integration over the reaction zone → Integral between T = ∞ und T = T b • When using the solution from the preheat zone, T is set equal to T . i b Preheat zone Y F,u Reaction zone F 22 Thermal Flame Theory (Summary) • Original derivation of the thermal flame theory of Zeldovich and Frank Kamenetzki not for reaction rate in the form of which is of first order with respect to both fuel and oxygen • Rather, different results for reaction rate of zeroth, first, and second order were derived Comparison with present result shows that first order valid for either very lean or very rich 23 Thermal Flame Theory (Summary) • Comparison with the following result: • Very rich/very lean mixture: reaction of first order  The component in shortage determines the conversion • Stoichiometric mixture: Reaction of second order  Both components are reaction rate determining 24 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Thermal flame theory • Fundamentals and mass • Flame thickness and flame time balances of combustion systems • Fourstep model for methaneair flames • Thermodynamics, flame temperature, and equilibrium • Threestep model for methaneair flames • Governing equations • Asymptotic structure of • Laminar premixed flames: stoichiometric methaneair 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 25 Flame Thickness and Flame Time • Thermal flame theory: • Burning velocity s : Eigenvalue, results from the solution of the one L dimensional balance equation • Assumptions:  One step reaction → Only one chemical time scale  Le = 1, thermal diffusivity equal to the mass diffusivity D • Solution for the burning velocity s : L links the parameters diffusivity and chemical time scale 26 Diffusivity and Chemical Time Scale • D: thermal diffusivity • Determined by r = r by l = l u b • Chemical time scale: • One or both of the reactants are consumed in the reaction zone • Square of ZeldovichNumber appears 27 Chemical Time Scale • Ze is of the order of 10 → Chemical time scale t by two orders of magnitude larger than the c chemical time scale which (apart from the density ratio r /r ) u b results from the reaction rate, e.g. for very lean mixtures f  1, as u the inverse of  t not determined by chemistry alone, additional influence of flame c structure • Definition of flame thickness: 28 Flame Time t F • Time, during which the flame front moves by one flame thickness • Comparison: • t is the flame time: c 29 Flame Thickness • Graphical determination of the flame thickness from the temperature profile:  Place tangent in the turning point of the profile  Intersections of the tangent with the horizontal lines at T and T u b  Length l at the abscissa F 30 Flame Thickness • In replacment of the left hand side by (T T )/ l and evaluation of the right hand b u F side at T = T yields: b 31 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Thermal flame theory • Fundamentals and mass • Flame thickness and flame time balances of combustion systems • Fourstep model for methaneair flames • Thermodynamics, flame temperature, and equilibrium • Threestep model for methaneair flames • Governing equations • Asymptotic structure of • Laminar premixed flames: stoichiometric methaneair 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 32 Flame structure analysis using multistep chemistry • Asymptotic description of premixed flames based on an assumed onestep reaction  Basic understanding of the flame structure when a large sensitivity to temperature was built into the model • There is no chemical basis for the onestep assumption  Results must be regarded with caution, especially Dependence of the burning velocity on pressure and composition Flammability and extinction limits • In contrast to simple analysis → Numerical calculations based on full and reduced mechanisms are able to predict these properties, but they contribute little to the understanding of the fundamental parameters that influence flame behavior 33 Mass Fraction Mass Fraction H Flame structure from multistep chemical kinetics • Structure of an unstretched premixed methane/air flame at standard conditions from numerical simulations 0.0003 0.14 0.12 0.00025 2000 2000 0.1 0.0002 CO 1500 2 1500 0.08 0.00015 0.06 1000 0.0001 1000 0.04 CO 5 5 10 0.02 500 500 0 0 T 1 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 4.5 5 y mm y mm 34 Temperature K Temperature KFlame structure from multistep chemical kinetics • Asymptotic description of stoichiometric methaneair flames based on four step reduced mechanism will be presented in this lecture • Similar asymptotic analysis was also carried out for lean, moderately rich, and rich methane flames (e.g. Seshadri 1991) • Description may, with some modifications, also serve as a model for other hydrocarbon flames • This will be shown by using analytical approximation formulas (Göttgens 1992) that are based on the asymptotic description of methane flames for flames of C H , C H , C H and C H in air 2 6 2 4 2 2 3 8 35 The FourStep Model for MethaneAir Flames • Systematically reduced mechanism using quasi steady state approximations • Non steady state components • Stable components CH , O , H O, CO , H , CO 4 2 2 2 2 o • H radical  Representing effect of radicals on flame structure  Competition between chain branching and chain breaking  Typically Hradical because of its importance in main radical reactions 36 The crossover (inner layer) temperature ≈ 1050 K • Most important chain branching and chain breaking reactions hydrogen and hydrocarbon combustion • Competition of 1f and 5f leads to 0 Crossover temperature T (aka: Inner layer temperature) 0 • T T : Chain termination, extinction 0 • T T : Chain branching, e.g. explosion 37 The FourStep Model for MethaneAir Flames • Global reaction III with the rate of reaction 1f describes chain branching • Global reaction IV with the rate of reaction 5f describes chain breaking 38 The FourStep Model for MethaneAir Flames • Global reaction I with the rate of reaction 38f describes fuel consumption Radical consuming because of CH consumption reaction 3 • Global reaction II with the rate of reaction 18f describes CO oxidation (water gas shift reaction) 39 The FourStep Model for MethaneAir Flames • The fourstep model for methane flames is in summary 40 The FourStep Model for MethaneAir Flames ≈ 1050 K • The principal rates governing these global reactions are • They correspond to the elementary reactions 0 Inner layer Temperature T 41 The FourStep Model for MethaneAir Flames • We neglect the influence of the other reactions here in order to make the algebraic description more tractable • Since OH and O appear in this formulation as reactants we need to express them in terms of the species in the fourstep mechanism by using the partial equilibrium assumption for the reaction such that where K and K are the equilibrium constants of reactions 2 and 3, respectively 2 3 42 The FourStep Model for MethaneAir Flames • This leads to the following reaction rates of the global steps IIV: which is explicit in terms of the concentrations of species appearing in the four step mechanism • The equilibrium constants in these rates are given by: 43 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Thermal flame theory • Fundamentals and mass • Flame thickness and flame time balances of combustion systems • Fourstep model for methaneair flames • Thermodynamics, flame temperature, and equilibrium • Threestep model for methaneair flames • Governing equations • Asymptotic structure of • Laminar premixed flames: stoichiometric methaneair 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 44 The ThreeStep Model for MethaneAir Flames • We now want to go one step further and assume steady state of the radical H • Adding reaction IV to I and III leads to the three steps with the first three rates given at the previous slide 45 The ThreeStep Model for MethaneAir Flames • H must now be determined from the steady state equation for H • Taking H balance in fourstep mechanism and setting H steady state leads to and • This may be written as where H based on partial equilibrium of reaction IV eq 46 The ThreeStep Model for MethaneAir Flames • The equation shows an interesting structure • At temperatures of 1400 K and above, the second term in the brackets is small while the ratio k / k is much larger than unity 11 1 CH /O must be much smaller than unity, if H is to remain real 4 2 • Equation cannot be valid in the preheat zone where second term is large • It also follows that H vanishes in the preheat zone, which is therefore chemically inert 47 Mass Fraction Mass Fraction H Flame Structure based on the fourstep mechanism ≈ 1050 K Results from detailed chemistry 0.0003 0.14 0.12 0.00025 2000 2000 0.1 0.0002 CO 1500 1500 2 0.08 0.00015 0.06 1000 0.0001 1000 0.04 CO 5 5 10 0.02 500 500 0 0 1 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 4.5 5 y mm y mm 48 Temperature K Temperature KCourse Overview Part I: Fundamentals and Laminar Flames • Introduction • Thermal flame theory • Fundamentals and mass • Flame thickness and flame time balances of combustion systems • Fourstep model for methaneair flames • Thermodynamics, flame temperature, and equilibrium • Threestep model for methaneair flames • Governing equations • Asymptotic structure of • Laminar premixed flames: stoichiometric methaneair 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 49 The Asymptotic Structure of Stoichiometric MethaneAir Flames • Further simplification couples CO and H leading to 2 twostep mechanism • It contains three layers 1. A chemically inert preheat zone of order 1 upstream 2. A thin inner layer of order d in which the fuel is consumed and the intermediates H and CO are formed according to the global step I'' 2 3. A thin oxidation layer of order e downstream where H and CO are 2 oxidized according to global step III'' 50 Details of the Asymptotic Analysis • At first the inner layer shall be analyzed • We will denote quantities at the inner layer with a subscript 0 but the inner 0 layer temperature as T • In this layer all concentrations except that of the fuel, which is depleted, may be assumed constant to leading order 51 Details of the Asymptotic Analysis • Introducing into leads to where the Damköhler number is 52 Details of the Asymptotic Analysis • The small parameter d was defined as • It denotes the ratio of the rate coefficients of reaction I and II • It hence describes the competition of these two reactions in producing and consuming Hradicals according to the global steps IV and I • Since it happens that the reaction rate k is typically smaller than k , and since 1 11 also X in the inner layer X 1, d ≈ 0.1 and sufficiently small for an O O 2 2 asymptotic expansion 53 Details of the Asymptotic Analysis • If d is small, since w must be real it follows from I the term in parenthesis that x must not exceed the value of d CH 4 • The flame structure shows that the fuel is of order 1 in the preheat zone but decreases rapidly towards the inner layer 54 Details of the Asymptotic Analysis • In the inner x is then of order d and one may introduce the scaling CH 4 and the stretched variable • Introducing these into it leads to the differential equation that governs the structure of the inner layer 55 Details of the Asymptotic Analysis • The downstream boundary condition of this equation is since reaction I is irreversible • The matching with the preheat zone should, as for the onestep asymptotic problem, provide the second boundary condition • The solution for the fuel concentration in the preheat zone is which leads to the expansion x = x around x = 0 CH 4 56 Details of the Asymptotic Analysis • It is shown in (Peters 1987), however, that the inner layer and the preheat zone are separated by an additional thin layer, the radical consumption layer • In this layer the steady state approximation for the Hradical breaks down • This layer occurs at y = 1, z = 1 in terms of the inner layer variables • Since the fuel is not consumed in this radical layer the slope of the fuel concentration is continuous and matching across this layer leads to 57 Details of the Asymptotic Analysis • With the boundary conditions and the equation can be integrated once to obtain the eigenvalue 58 Details of the Asymptotic Analysis • With 0 one could now determine the burning velocity s if the temperature T and all L other properties at the inner layer were known • In order to determine these, the structure of the oxidation layer also must be resolved • In the oxidation layer x = 0 and therefore w = 0 CH I 4 • The temperature varies only slowly in this layer and since the activation energy of k is small, temperature variations may be neglected 5 59 Details of the Asymptotic Analysis • Since most of the chemical activity takes place in the vicinity of the inner layer, all properties shall be evaluated at x = 0 • Choosing x as the dependent variable in the oxidation layer and scaling it in H 2 terms of a new variable z as • One may use the coupling relations to show that the downstream boundary conditions are satisfied by 60 Details of the Asymptotic Analysis • In these expansions e is the small parameter related to the thickness of the oxidation layer. • Introducing and into leads to where the Damköhler number of reaction III is defined as 61 Details of the Asymptotic Analysis • The concentration of the third body in reaction 5 may be determined approximately by using the third body efficiencies evaluated at the burnt gas conditions • This leads to which introduces a pressure dependence of Da and will finally determine the III pressure dependence of the burning velocity. 62 Details of the Asymptotic Analysis • Introduction of a stretched coordinate then leads with w = 0 from I to the governing equation of the oxidation layer • This suggests the definition • It turns out that for p ≥ 1 atm e is smaller than unity but typically larger than d 63 Details of the Asymptotic Analysis • Even though d is not very small, we will consider it as small enough to justify an asymptotic description of the oxidation layer • The downstream boundary condition of equation is since reaction III is irreversible • The upstream boundary condition must be determined from jump conditions across the inner layer 64 Details of the Asymptotic Analysis • Since the fuel is depleted and H is formed in the inner layer following reaction 2 I'', the stoichiometry of this reaction also determines the change of slopes of the H in comparison of those of the fuel 2 • This is written as • Since the thickness of the preheat zone is of order 1 and that of the oxidation layer of order e the upstream slope of the H concentration can be neglected 2 compared to the downstream slope 65 Details of the Asymptotic Analysis • It then follows with and that the upstream boundary condition of reads Then the solution is with 66 Details of the Asymptotic Analysis • The profile shows a very slow decrease of z towards h   • This may explain why in numerically and experimentally obtained concentration and temperature profiles the downstream conditions are approached only very far behind the flame 67 An Analytic Expression for the Burning Velocity • The result may now be used in and to determine the quantities required in and thereby the burning velocity s L 68 An Analytic Expression for the Burning Velocity • By dividing by one can eliminate s and obtain a relation of the form L 3 • Here the universal gas constant must be used as = 82.05 atm cm /mol/K in order to be consistent with the units of the reaction rates and the pressure 69 An Analytic Expression for the Burning Velocity • The equation shows that with the rate coefficients fixed, inner layer temperature is function of the pressure only • It does not depend on the preheat temperature, the dilution of the fuel concentration in the unburnt mixture and thereby the adiabatic flame temperature 70 An Analytic Expression for the Burning Velocity • After some algebraic manipulations the expression for the burning velocity reads where and 0 were used to relate e to the difference between T and T b 71 Results of the Asymptotic Analysis Asymptotic Analysis leads to analytic expression for laminar burning velocity • Pressure obtained from , • For an undiluted flame with T u = 300 K and p = 1 atm, 0, determining T one obtains a laminar burning velocity of 54 cm/s for stoichiometric methane flames 72 Results of the Asymptotic Analysis • This value is satisfactory in view of the many approximations that were made and the few kinetic rates that were retained • In fact, it is seen from and that only the rates of reactions 1, 5, and 11 influence the burning velocity in this approximation 73 Results of the Asymptotic Analysis • A further consequence of equation 0 is that the burning velocity vanishes as T reaches T b 0 • With T = 2320 K, T b reaches T when the b pressure is larger than approximately 20 atm 74 Results of the Asymptotic Analysis • Different values of T would have been obtained for a diluted or preheated b flame 0 • The fact that at a fixed pressure T is fixed by the ratio of rate coefficients points towards the possibility to explain flammability limits at least in terms of dilution for stoichiometric flames If the amount of fuel is so low that in the unburnt mixture the corresponding adiabatic flame temperature is lower than T , a 0 premixed flame cannot be established 75 Detail: Relation to the Activation Energy of the Onestep Model • Using the burning velocity expression from the thermal flame theory one may plot the burning velocity in an Arrhenius diagram over 1/T b 76 Detail: Relation to the Activation Energy of the Onestep Model • Then in the limit of a large activation energy, the slope in this diagram is given by or • Applying this form to burning velocity from 2step mechanism 0 with T fixed leads to 77 Detail: Relation to the Activation Energy of the Onestep Model • Since the second of the terms is much smaller then the first, with , where z is the scaled hydrogen mass fraction in 0 the inner layer, 0 and when T approaches T and e is small, one obtains b • Therefore the Zeldovich number introduced in the previous lecture may be expressed as 78 Relation to the Activation Energy of the Onestep Model • Onestep model • Reaction zone thickness was of order of the inverse Zeldovich number • Twostep model for methane flames • Oxidation layer thickness of order of the inverse Zeldovich number Oxidation layer plays similar role in hydrocarbon flames as reaction zone in onestep asymptotics 79 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Thermal flame theory • Fundamentals and mass • Flame thickness and flame time balances of combustion systems • Fourstep model for methaneair flames • Thermodynamics, flame temperature, and equilibrium • Threestep model for methaneair flames • Governing equations • Asymptotic structure of • Laminar premixed flames: stoichiometric methaneair 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 80 Application: Analytic Approximations of Burning Velocities for Lean CH , C H , C H , C H , and C H Flames 4 2 6 2 4 2 2 3 8 • Burning velocity expression presented may be generalized by writing an approximation formula for burning velocities as and 0 0 where the functions A(T ) and P(T ) are determined by fitting numerical or experimental data and the values m = 1/2 and n = 2 would correspond to the previous expressions for premixed methane flames 81 Analytic Approximations of Burning Velocities for Lean CH , C H , C H , C H , and C H Flames 4 2 6 2 4 2 2 3 8 • assumes that the inner layer temperature is a function of pressure only, and it does not depend, for instance, on the equivalence ratio • This is a fairly crude approximation as may be seen when inner layer temperatures obtained from asymptotic analysis (Seshadri 1991) are plotted together with the adiabatic temperatures as a function of the equivalence ratio 82 Analytic Approximations of Burning Velocities for Lean CH , C H , C H , C H , and C H Flames 4 2 6 2 4 2 2 3 8 • If the structure of any other hydrocarbon fuel is similar to that of methane, these exponents should not differ very much from these numbers 0 0 • Since A(T ) and P(T ) contain essentially the temperature dependence due to rate coefficients, we express them in Arrhenius form • This concept was tested by Göttgens (1992) • Basis of approximation was a data set of 197, 223, 252, 248, and 215 premixed flames for CH , C H , C H , C H and C H , in the range between 4 2 6 2 4 2 2 3 8 p = 1 atm and 40 atm, T between 298 K and 800 K, and the fuelair u equivalence ratio between f = 0.4 and 1.0 83 Analytic Approximations of Burning Velocities for Lean CH , C H , C H , C H , and C H Flames 4 2 6 2 4 2 2 3 8 • A nonlinear approximation procedure was employed, yielding the following values for the coefficients: • The approximation was surprisingly the best for C H , yielding a standard 2 2 deviation for s of 2.3, followed by C H with 3.2, C H and C H with 6.2, L 2 4 2 6 3 8 and CH with 7.4 4 84 Analytic Approximations of Burning Velocities for Lean CH , C H , C H , C H , and C H Flames 4 2 6 2 4 2 2 3 8 • These deviations may be considered extremely small in view of the fact that such a large range of equivalence ratios, pressures and preheat temperatures has been covered with an approximation formula containing only six coefficients • A closer look at the exponents m and n shows that m is close to 1/2 for CH and 4 C H , but close to unity for C H and C H 3 8 2 2 2 4 • This suggests that the asymptotic model for these flames should differ from the one for CH in some important details 4 • The exponent m lies around 2.5 and thereby sufficiently close to 2 for all fuels 85 Analytic Approximations of Burning Velocities for Lean CH , C H , C H , C H , and C H Flames 4 2 6 2 4 2 2 3 8 • Burning velocities for methane calculated from and are plotted as a function of equivalence ratio for different pressures at T = 298 K and u compared with the values obtained from the numerical computations. • Generally the largest derivations from the numerical computations occur around f = 1 86 Analytic Approximations of Burning Velocities for Lean CH , C H , C H , C H , and C H Flames 4 2 6 2 4 2 2 3 8 • Burning velocities for methane calculated from and • The pressure and unburnt temperature variation of s L at stoichiometric mixture are plotted for propane 87 Example • From the approximation calculate in comparison with the activation energy that describes the change of the reaction rate as function of the change in T b 0 • Thereby T and T should be considered constant u 88 Solution • If one writes approximately as and logarithmizes this expression: one can determine the activation energy by differentiation with respect to 1/T b 89 Solution • This leads to • Using this in for r = const, it follows u • Therefore one obtains for the Zeldovich number Ze 90 Solution • Here, following 0 T is only dependent on pressure, while T follows b depends both on T and on the fuelair ratio f = 1/l u • If the difference T T is small compared with T T , the second term in the b 0 b u parenthesis can be neglected 91 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Thermal flame theory • Fundamentals and mass • Flame thickness and flame time balances of combustion systems • Fourstep model for methaneair flames • Thermodynamics, flame temperature, and equilibrium • Threestep model for methaneair flames • Governing equations • Asymptotic structure of • Laminar premixed flames: stoichiometric methaneair 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 92 Application: Flammability limits • Lean and rich flammability limits are function of temperature and pressure • For lean mixtures (RHS of the diagram), flammability limits of hydrogen extend to much larger values than for methane • This shows that hydrogen leakage may cause greater safety hazards than, for instance, hydrocarbons, which have flammability limits close to those of methane 93 Kinetically determined flammability limit 0 • Temperature T of the inner layer  Corresponds to T in the approximation equation: 0 0 • T : „crossover“temperature between chain termination and chain branching → kinetically determined 94 Kinetically determined lean flammability limit • Approximation equation:  T = T : Burning velocity = 0 b 0 • From approximation of the coefficients:  T depends on pressure but not on the fuel mass fraction 0 • Condition T = T : Decreasing the fuel mass fraction in the mixture → T decreases b 0 b • Corresponds to approaching the lean flammability limit • Fuel mass fraction too low: Inner „crossover“ temperature T is not reached 0 No chain branching Extinction 95 Flammability limit from coupling function • Coupling function yields a relation for Y : F,u 0  with T = T b  complete combustion (Y = 0) F,b • Eliminate values of the mass fraction with coupling equation for stoichiometric mixture Mass fraction of the fuel at the lean flammability limit compared to the stoichiometric mixture • Example: Methaneairflame:  T = 300 K , p = 1 bar u 0  T = 1219 K , T = 2229 K st  Corresponds to l = 2,16  Upper value for the lean flammability limit 96 Flammability limits of real flames • Real situations:  Flame extinction occurs sooner • Iterative calculation of the limit Y from s B,u L • Increasing temperature  Mole fraction decreases  Region of flammable mixture broadens 0 • T = T : Lower value of the mole b fraction → Kinetically determined 97 Theoretical explanation of the lean flammability limit • Thermal Flame Theory: No flammability limit • Exponential dependence of the laminar burning velocity on the temperature in the burned mixture: Laminar burning velocity takes very low values with decreasing T but will b never become zero • Flame propagation can be disrupted due to heat loss effects 98 Summary Part I: Fundamentals and Laminar Flames • Introduction • Thermal flame theory • Fundamentals and mass • Flame thickness and flame time balances of combustion systems • Fourstep model for methaneair flames • Thermodynamics, flame temperature, and equilibrium • Threestep model for methaneair flames • Governing equations • Asymptotic structure of • Laminar premixed flames: stoichiometric methaneair 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 99