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Laminar Premixed Flames: Kinematics and Burning Velocity

Laminar Premixed Flames: Kinematics and Burning Velocity
Laminar Premixed Flames: Kinematics and Burning Velocity 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 • Introduction • Fundamentals and mass • Kinematic balance for steady balances of combustion systems oblique flames • Thermodynamics, flame • Laminar burning velocity temperature, and equilibrium • Field equation for the flame • Governing equations position • Laminar premixed flames: • Flame stretch and curvature Kinematics and Burning Velocity • Thermaldiffusive flame instability • Laminar premixed flames: • Hydrodynamic flame instability Flame structure • Laminar diffusion flames 2 Laminar Premixed Flames • Premixed combustion used in combustion devices when high heat release rates are desired  Small devices  Low residence times • Examples:  SI engine  Stationary gas turbines • Advantage  Lean combustion possible  Smokefree combustion  Low NO x • Disadvantage: Danger of  Explosions  Combustion instabilities  Largescale industrial furnaces and aircraft engines are typically nonpremixed 3 Premixed Flames • Premixed flame: Blue or bluegreen by chemiluminescence of excited radicals, o o such as C and CH 2 • Diffusion flames: Yellow due to soot radiation Turbulent Laminar Premixed Flame Bunsen Flame (Dunn et al.) 4 Flame Structure of Premixed Laminar Flames • Fuel and oxidizer are convected from upstream Cut through with the burning velocity s L flame • Fuel and air diffuse into the reaction zone • Mixture heated up by heat conduction from the burnt gases • Fuel consumption, radical production, and oxidation when inner layer temperature is reached • Increase temperature and gradients • Fuel is entirely depleted • Remaining oxygen is convected downstream 5 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Introduction • Fundamentals and mass • Kinematic balance for steady balances of combustion systems oblique flames • Thermodynamics, flame • Laminar burning velocity temperature, and equilibrium • Field equation for the flame • Governing equations position • Laminar premixed flames: • Flame stretch and curvature Kinematics and Burning Velocity • Thermaldiffusive flame instability • Laminar premixed flames: • Hydrodynamic flame instability Flame structure • Laminar diffusion flames 6 Premixed Flame in a Bunsen Burner • Fuel enters the Bunsen tube with high momentum through a small orifice • High momentum  underpressure  air entrainment into Bunsen tube • Premixing of fuel and air in the Bunsen tube • At tube exit: homogeneous, premixed fuel/air mixture, which can and should() be ignited 7 Kinematic Balance for Steady Oblique Flame • In steady state, flame forms Bunsen cone • Velocity component normal to flame front is locally equal to the propagation velocity of the flame front  Burning velocity 8 Kinematic Balance for Steady Oblique Flame • Laminar burning velocity s : Velocity of L,u the flame normal to the flame front and relative to the unburnt mixture (index ‘u’) • Can principally be experimentally determined with the Bunsen burner • Need to measure Velocity of mixture at Bunsen tube exit Bunsen cone angle α 9 Kinematic Balance for Steady Oblique Flame • Splitting of the tube exit velocity in components normal and tangential to the flame • Kinematic balance yields relation unburnt gas velocity and flame propagation velocity • For laminar flows: 10 Kinematic Balance for Steady Oblique Flame • Flame front: • Large temperature increase • Pressure almost constant  Density decreases drastically • Mass balance normal to the flame front: • Normal velocity component increases through flame front • Momentum balance in tangential direction:  Deflection of the streamlines away from the flame Laminar Bunsen flame (Mungal et al.) 11 Burning Velocity at the Flame Tip • Tip of the Bunsen cone Symmetry line Burning velocity equal to velocity in unburnt mixture Here: Burning velocity = normal component, tangential component = 0 Laminar Bunsen flame (Mungal et al.)  Burning velocity at the tip by a factor 1/sin(α) larger than burning velocity through oblique part of the cone 12 Burning velocity at the flame tip • Explanation: Strong curvature of the flame front at the tip  Increased preheating In addition to heat conduction normal to the flame front preheating by the lateral parts of the flame front Laminar Bunsen flame (Mungal et al.) • Effect of nonunity Lewis numbers  Explanation of difference between lean hydrogen and lean hydrocarbon flames 13 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Introduction • Fundamentals and mass • Kinematic balance for steady balances of combustion systems oblique flames • Thermodynamics, flame • Laminar burning velocity temperature, and equilibrium • Field equation for the flame • Governing equations position • Laminar premixed flames: • Flame stretch and curvature Kinematics and Burning Velocity • Thermaldiffusive flame instability • Laminar premixed flames: • Hydrodynamic flame instability Flame structure • Laminar diffusion flames 14 Measuring the laminar burning velocity • Spherical constant volume combustion vessel Flame initiated by a central spark Spherical propagation of a flame Measurements of radial flame propagation velocity dr /dt f • Kinematic relation for flame displacement speed • Flame front position and displacement speed are unsteady • Pressure increase negligible as long as volume of burnt mixture small relative to total volume • Influence of curvature 15 Measuring the laminar burning velocity 16 Flame front velocity in a spherical combustion vessel • Velocity relative to flame front is the burning velocity • Different in burnt and unburnt region dr /dt f • From kinematic relation v u • Velocity on the unburnt side (relative to the flame front) • Burnt side of the front • Spherical propagation: Due to symmetry, flow velocity in the burnt gas is zero • Mass balance yields: 17 Flame front velocity in a spherical combustion vessel • From mass balance and kinematic relation follows • Flow velocity on the unburnt side of the front  Flow of the unburnt mixture induced by the expansion of the gases behind the flame front • Measurements of the flame front velocity dr /dt f  Burning velocity s : L,u 18 Relation between s and s L,u L,b • Burning velocity s defined with respect to the unburnt mixture L,u • Another burning velocity s can be defined with respect to the L,b burnt mixture • Continuity yields the relation: • In the following, we will usually consider the burning velocity with respect to the unburnt s = s L L,u 19 Flat Flame Burner and Flame Structure • Onedimensional flame • Stabilization by heat losses to burner • In theory, velocity could be increased until heat losses vanish, then  unstretched  u = s u L • Analysis of flame structure of flat flames Measurements of temperature and species concentration profiles 20 The general case with multistep chemical kinetics • Laminar burning velocity s can be calculated by solving governing conservation L equations for the overall mass, species, and temperature (low Mach limit) • Continuity • Species • Energy 21 The general case with multistep chemical kinetics • Continuity equation may be integrated once to yield • Burning velocity is eigenvalue, which must be determined as part of the solution • System of equations may be solved numerically with Appropriate upstream boundary conditions Zero gradient boundary conditions downstream 22 The general case with multistep chemical kinetics • Example: Calculations of the burning velocity of premixed methaneair flames • Mechanism that contains only C hydrocarbons 1  s underpredicted L • Including C mechanism 2 Mauss 1993  Better agreement 23 The general case with multistep chemical kinetics • Example: Burning velocities of propane flames taken from Kennel (1993) • s typically decreases with increasing pressure but increases with increasing L preheat temperature 24 Burning Velocity • Burning velocity is fundamental property of a premixed flame • Can be used to determine flame dynamics • Depends on thermochemical parameters of the premixed gas ahead of flame only But:  For Bunsen flame, the condition of a constant burning velocity is violated at the tip of the flame  Curvature must be taken into account Next • We will first calculate flame shapes • Then we will consider external influences that locally change the burning velocity and discuss the response of the flame to these disturbances 25 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Introduction • Fundamentals and mass • Kinematic balance for steady balances of combustion systems oblique flames • Thermodynamics, flame • Laminar burning velocity temperature, and equilibrium • Field equation for the flame • Governing equations position • Laminar premixed flames: • Flame stretch and curvature Kinematics and Burning Velocity • Thermaldiffusive flame instability • Laminar premixed flames: • Hydrodynamic flame instability Flame structure • Laminar diffusion flames 26 A Field Equation Describing the Flame Position • Kinematic relation between • Displacement velocity • Flow velocity • Burning velocity • May be generalized by introducing vector n normal to the flame where x is the vector describing f the flame position, dx /dt the flame propagation f velocity, and v the velocity vector 27 A Field Equation Describing the Flame Position • Normal vector points towards the unburnt mixture and is given by where G(x,t) can be identified as a scalar field whose level surface represents the flame surface and G is arbitrary 0 • The flame contour G(x,t) = G divides physical field into two regions, where 0 G G is the region of burnt gas and G G that of the unburnt mixture 0 0 28 A Field Equation Describing the Flame Position • Differentiating G(x,t) = G with respect to t at G = G gives 0 0 • Introducing leads to • Level set equation for the propagating flame follows using as 29 A Field Equation Describing the Flame Position • Burning velocity s is defined w.r.t. the unburnt mixture L  Flow velocity v is defined as the conditioned velocity field in the unburnt mixture ahead of the flame • For a constant value of s the solution of L, is nonunique, and cusps will form where different parts of the flame intersect • Even an originally smooth undulated front in a quiescent flow will form cusps and eventually become flatter with time • This is called Huygens' principle 30 Exercise: Slot Burner • A closed form solution of the Gequation can be obtained for the case of a slot burner with a constant exit velocity u for premixed combustion, • This is the twodimensional planar version of the axisymmetric Bunsen burner. • The Gequation takes the form 31 Exercise: Slot Burner • With the ansatz and G = 0 one obtains 0 leading to • As the flame is attached at x = 0, y = ± b/2, where G = 0, this leads to the solution 32 Exercise: Slot Burner The flame tip lies with y=0, G = 0 at and the flame angle a is given by With it follows that , which is equivalent to . This solution shows a cusp at the flame tip x = x , y = 0. In order to obtain a F0 rounded flame tip, one has to take modifications of the burning velocity due to flame curvature into account. This leads to the concept of flame stretch. 33 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Introduction • Fundamentals and mass • Kinematic balance for steady balances of combustion systems oblique flames • Thermodynamics, flame • Laminar burning velocity temperature, and equilibrium • Field equation for the flame • Governing equations position • Laminar premixed flames: • Flame stretch and curvature Kinematics and Burning Velocity • Thermaldiffusive flame instability • Laminar premixed flames: • Hydrodynamic flame instability Flame structure • Laminar diffusion flames 34 Flame stretch • Flame stretch consists of two contributions: • Flame curvature • Flow divergence or strain • For onestep large activation energy reaction and with the assumption of constant properties, the burning velocity s is modified by these two effects as L 0 • s is the burning velocity for an unstretched flame L • is the Markstein length 35 Flame stretch • The flame curvature k is defined as which may be transformed as • The Markstein length appearing in is of same order of magnitude and proportional to laminar flame thickness • Ratio is called Markstein number 36 Markstein length • With assumptions: • Onestep reaction with a large activation energy • Constant transport properties and heat capacity c p  Markstein length with respect to the unburnt mixture Unstretched laminar burning velocity • Markstein length  Determined experimentally  Determined by asymptotic analysis Density ratio ZeldovichNumber LewisNumber 37 Markstein length • Markstein length • Derived by Clavin and Williams (1982) and Matalon and Matkowsky (1982) • is the Zeldovich number, where E is the activation energy, the universal gas constant, and Le the Lewis number of the deficient reactant • Different expression can be derived, if both s and are defined with respect to L the burnt gas cf. Clavin, 1985 38 Example: Effect of Flame Curvature • We want to explore the influence of curvature on the burning velocity for the case of a spherical propagating flame • Flow velocity is zero in the burnt gas  Formulate the Gequation with respect to the burnt gas: where r (t) is the radial flame position f 0 • The burning velocity is then s and the Markstein length is that with respect L,b to the burnt gas . • Here, we assume to avoid complications associated with thermo diffusive instabilities 39 Example: Effect of Flame Curvature • In a spherical coordinate system, the Gequation reads where the entire term in round brackets represents the curvature in spherical coordinates • We introduce the ansatz to obtain at the flame front r=r f • This equation may also be found in Clavin (1985) 40 Example: Effect of Flame Curvature • This equation reduces to for • It may be integrated to obtain where the initial radius at t=0 is denoted by r f,0 • This expression has no meaningful solutions for , indicating that there needs to be a minimum initial flame kernel for flame propagation to take off • It should be recalled that is only valid if the product • For curvature corrections are important at early times only 41 Effects of curvature and strain on laminar burning velocity Curvature Effect on Laminar Burning Strain Effect on Laminar Burning Velocity from Numerical Simulations Velocity from Experiments and Theory Laminar premixed stoichiometric methane/air counterflow flames Laminar premixed stoichiometric f = 0.8 methane/air spherically expanding flames Note: s ≈ s /7 L,u L,b f = 1 42 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Introduction • Fundamentals and mass • Kinematic balance for steady balances of combustion systems oblique flames • Thermodynamics, flame • Laminar burning velocity temperature, and equilibrium • Field equation for the flame • Governing equations position • Laminar premixed flames: • Flame stretch and curvature Kinematics and Burning Velocity • Thermaldiffusive flame instability • Laminar premixed flames: • Hydrodynamic flame instability Flame structure • Laminar diffusion flames 43 Flame Instabilities: Thermaldiffusive instability Effect of Curvature Effect of stretch Unstretched laminar burning velocity Unburnt Burnt 44 Flame Instabilities: Thermaldiffusive instability Unstretched laminar burning velocity Unburnt Burnt 45 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Introduction • Fundamentals and mass • Kinematic balance for steady balances of combustion systems oblique flames • Thermodynamics, flame • Laminar burning velocity temperature, and equilibrium • Field equation for the flame • Governing equations position • Laminar premixed flames: • Flame stretch and curvature Kinematics and Burning Velocity • Thermaldiffusive flame instability • Laminar premixed flames: • Hydrodynamic flame instability Flame structure • Laminar diffusion flames 46 Flame Instabilities: Hydrodynamic Instability • Illustration of the hydrodynamic instability of a slightly undulated flame • Gas expansion in the flame front leads to a deflection of a stream line that enters the front at an angle • A stream tube with crosssectional area A and upstream flow velocity u 0  widens due to flow divergence ahead of the flame 47 Flame Instabilities: Hydrodynamic Instability • Expansion at the front induces a flow component normal to the flame contour • As the stream lines cross the front they are deflected • At large distances from front, stream lines are parallel again, but downstream velocity is • At a cross section A , where density is still equal to r , by continuity flow 1 u velocity becomes 48 Flame Instabilities: Hydrodynamic Instability • The unperturbed flame propagates with normal to itself • Burning velocity is larger than u , 1 flame propagates upstream and thereby enhances the initial perturbation • Analysis can be performed with following simplifications • Viscosity, gravity and compressibility in the burnt and unburnt gas are neglected • Density is discontinuous at the flame front • The influence of the flame curvature on the burning velocity is retained, flame stretch due to flow divergence is neglected 49 Flame Instabilities: Hydrodynamic Instability • Analysis results in dispersion relation where s is the nondimensional growth rate of the perturbation r is density ratio and k the wave number • Perturbation grows exponentially in time only for a certain wavenumber range 0 k k with 50 Flame Instabilities: Hydrodynamic Instability • Without influence of curvature ( ), flame is unconditionally unstable • For perturbations at wave numbers k k, a planar flame of infinitively small thickness, described as a discontinuity in density, velocity and pressure is unconditionally stable • Influence of front curvature on burning velocity • As one would expect on the basis of simple thermal theories of flame propagation, burning velocity increases when flame front is concave and decreases when it is convex towards unburnt gas, so that initial perturbations become smoother 51 Details of the Analysis for Hydrodynamic Instability • The burning velocity is given by • Reference values for length, time, density, pressure: • Introduce the density rate: • Dimensionless variables: 52 Details of the Analysis for Hydrodynamic Instability • The nondimensional governing equations are then (with the asterisks removed) where r = 1 and r= r in the unburnt and burnt mixture respectively. u • If G is a measure of the distance to the flame front, the Gfield is described by: 53 Details of the Analysis for Hydrodynamic Instability • With equations the normal vector n and the normal propagation velocity then are 54 Details of the Analysis for Hydrodynamic Instability • Due to the discontinuity in density at the flame front, the Euler equations are only valid on either side of the front, but do not hold across it. • Therefore jump conditions for mass and momentum conservation across the discontinuity are introduced Williams85,p. 16: • The subscripts + and refer to the burnt and the unburnt gas and denote the properties immediately downstream and upstream of the flame front. 55 Details of the Analysis for Hydrodynamic Instability • In terms of the u and v components the jump conditions read • Under the assumption of small perturbations of the front, with e 1 the unknowns are expanded as 56 Details of the Analysis for Hydrodynamic Instability • Jump conditions to leading order and to first order where the leading order mass flux has been set equal to one: 57 Details of the Analysis for Hydrodynamic Instability • With the coordinate transformation we fix the discontinuity at x = 0. • To first order the equations for the perturbed quantities on both sides of the flame front now read where r = 1 for x 0 (unburnt gas) and r = r for x 0 (burnt gas) is to be used. • In case of instability perturbations which are initially periodic in the hdirection and vanish for x  ±  would increase with time. 58 Details of the Analysis for Hydrodynamic Instability • Since the system is linear, the solution may be written as where s is the nondimensional growth rate, k the nondimensional wave number and i the imaginary unit. • Introducing this into the first order equations the linear system may be written as • The matrix A is given by 59 Details of the Analysis for Hydrodynamic Instability • The eigenvalues of A are obtained by setting det(A) = 0. • This leads to the characteristic equation • Here again U = 1/r, r = r for x 0 and U = 1, r = 1 for x 0. • There are three solutions to the characteristic equation for the eigenvalues a , j = 1,2,3. j • Positive values of a satisfy the upstream (x 0) and negative values the j downstream (x 0) boundary conditions of the Euler equations. 60 Details of the Analysis for Hydrodynamic Instability • Therefore • Introducing the eigenvalues into again, the corresponding eigenvectors w , j = 1,2,3 are calculated to 0,j 61 Details of the Analysis for Hydrodynamic Instability • In terms of the original unknowns u, v and the solution is now • For the perturbation f (h, t) the form will be introduced. 62 Details of the Analysis for Hydrodynamic Instability • Inserting and into the nondimensional Gequation satisfies to leading order with and x = 0 , x = 0 respectively. + 63 Details of the Analysis for Hydrodynamic Instability • This leads to first order to • With the jump conditions • can be written as 64 Details of the Analysis for Hydrodynamic Instability • The system then reads 65 Details of the Analysis for Hydrodynamic Instability • Since equation is linear dependent from equations it is dropped and the equations and remain for the determination of a, b, c and s(k). 66 Details of the Analysis for Hydrodynamic Instability • Dividing all equations by one obtains four equations for • The elimination of the first three unknown yields the equation • The solution may be written in terms of dimensional quantities as • Here only the positive root has been taken, since it refers to possible solutions with exponential growing amplitudes. 67 Details of the Analysis for Hydrodynamic Instability The relation is the dispersion relation which shows that the perturbation f grows exponentially in time only for a certain wavenumber range 0 k k . Here k is the wave number of which j = 0 in which leads to 68 Exercise • Under the assumption of a constant burning velocity s = s the linear stability L L0 analysis leads to the following dispersion relation • Validate this expression by inserting • What is the physical meaning of this result • What effect has the front curvature on the flame front stability 69 Exercise Solution • The dispersion relation for constant burning velocity s = s , L L0 shows that the perturbation F grows exponentially in time for all wave numbers. • The growth s is proportional to the wave number k and always positive since the density rate r is less than unity. • This means that a plane flame front with constant burning velocity is unstable to any perturbation. 70 Exercise • The front curvature has a stabilizing effect on the flame front stability. • As it is shown in the last section, the linear stability analysis for a burning velocity with the curvature effect retained leads to instability of the front only for the wave number range whereas the front is stable to all perturbations with k k. 71 Summary Part I: Fundamentals and Laminar Flames • Introduction • Introduction • Fundamentals and mass • Kinematic balance for steady balances of combustion systems oblique flames • Thermodynamics, flame • Laminar burning velocity temperature, and equilibrium • Field equation for the flame • Governing equations position • Laminar premixed flames: • Flame stretch and curvature Kinematics and Burning Velocity • Thermaldiffusive flame instability • Laminar premixed flames: • Hydrodynamic flame instability Flame structure • Laminar diffusion flames 72
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