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Laminar Diffusion Flames

Laminar Diffusion Flames
Laminar Diffusion Flames 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 • Fundamentals and mass balances of combustion systems • Introduction • Thermodynamics, flame • Counterflow diffusion flame temperature, and equilibrium • Flamelet structure of diffusion flames • Governing equations • FlameMaster flame calculator • Laminar premixed flames: Kinematics and Burning Velocity • Single droplet combustion • Laminar premixed flames: Flame structure • Laminar diffusion flames 2 Laminar diffusion flames • Seperate feeding of fuel and oxidizer into the combustion chamber  Diesel engine  Jet engine • In the combustion chamber:  Mixing Injection and  Subsequently combustion combustion in a • Mixing: Convection and diffusion diesel engine  On a molecular level  (locally) stoichiometric mixture • Simple example for a diffusion flame: Candle flame  Paraffin vaporizes at the wick → diffuses into the surrounding air • Simultaneously: Air flows towards the flame due to free convection and forms a mixture with the vaporized paraffin 3 Candle flame yellow region (soot particles) dark region with thin blue layer (chemiluminescence) vaporized paraffin air air • In a first approximation, combustion takes place at locations, where the concentrations of oxygen and fuel prevail in stoichiometric conditions. 4 Comparison of laminar premixed and diffusion flames Temperature Fuel Fuel Oxidizer Reaction rate Reaction rate Temperature Oxidizer Structure of a diffusion flame (schematic) Structure of a premixed flame (schematic) 5 Soot in candle flames • Soot particles  Formation in fuel rich regions of the flame  Transported to lean regions through the surface of stoichiometric mixture  In the oxygen containing ambient: Combustion of the soot particles • Sooting flame: Residence time of the soot particles in the region of oxidizing ambient and high temperatures too short to burn all particles 6 Timescales • Considering the relative times required for  Convection and diffusion  Proceeding reactions • For technical combustion processes in diffusion flames:  Characteristic times of convection and diffusion are approximately of same order of magnitude  Characteristic times of chemical reactions much smaller • Limit of fast chemical reactions  Mixing is the slowest and therefore rate determining process → “mixed = burnt” 7 The mixture fraction • Mixture fraction: • Stoichiometric mixture fraction: • Relation with equivalence ratio  Pure oxidizer (f = 0): Z = 0  Pure fuel (f = ∞): Z = 1 8 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Fundamentals and mass balances of combustion systems • Introduction • Thermodynamics, flame • Counterflow diffusion flame temperature, and equilibrium • Flamelet structure of diffusion flames • Governing equations • FlameMaster flame calculator • Laminar premixed flames: Kinematics and Burning Velocity • Single droplet combustion • Laminar premixed flames: Flame structure • Laminar diffusion flames 9 Counterflow Diffusion flame • Onedimensional similarity solution • Strain appears as parameter  Da • Used for  Studying flame structure  Studying chemistry in diffusion flames  Study interaction of flow and chemistry 10 Counterflow diffusion flame: Governing Equations • Continuity • X – Momentum • Energy 11 Counterflow diffusion flame: Similarity solution • Three assumptions reduce systems of equation to 1D 1. Similarity assumption for velocity 2. Similarity assumption 3. Mass fractions and temperature have no radial dependence close to centerline 12 Counterflow diffusion flame: Similarity solution • This results in • with boundary conditions 13 Counterflow diffusion flame: Similarity solution • Alternatively, potential flow boundary conditions can be used at y  ±∞ instead of nozzles • With definition of strain rate the similarity coordinate h the nondimensional stream function f defined by and the ChapmanRubesin parameter the 1D similarity solution can be derived 14 Counterflow diffusion flame: Similarity solution • Potential flow similarity solution • With Dirichlet boundary conditions for mass fractions and temperature and where the velocities are obtained from 15 Structure of nonpremixed laminar flames Temperature for methane/air counterflow diffusion flames Structure of nonpremixed laminar flames Maximum flame temperature for methane/air counterflow diffusion flames Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Fundamentals and mass balances of combustion systems • Introduction • Thermodynamics, flame • Counterflow diffusion flame temperature, and equilibrium • Flamelet structure of diffusion flames • Governing equations • FlameMaster flame calculator • Laminar premixed flames: Kinematics and Burning Velocity • Single droplet combustion • Laminar premixed flames: Flame structure • Laminar diffusion flames 18 Theoretical description of diffusion flames • Assumption of fast chemical reactions  Without details of the chemical kinetics  Global properties, e.g. flame length • If characteristic timescales of the flow and the reaction are of same order of magnitude:  Chemical reaction processes have to be considered explicitly  Liftoff and extinction of diffusion flames  Formation of pollutants • Flamelet formulation for nonpremixed combustion • Mixture fraction as independent coordinate for all reacting scalars, • Asymptotic approximation in the limit of sufficiently fast chemistry to one dimensional equations for reaction zone 19 Flamelet structure of a diffusion flame • Assumptions: Equal diffusivities of chemical species and temperature • The balance equation for mixture fraction, temperature and species read • Low Mach number limit • Zero spatial pressure gradients • Temporal pressure change is retained 20 Flamelet structure of a diffusion flame • Balance equation for the mixture fraction • No chemical source term, since elements are conserved in chemical reactions • We assume the mixture fraction Z to be given in the flow field as a function of space and time: Z = Z(x ,t) a 21 Flamelet structure of a diffusion flame • Surface of the stoichiometric mixture: • If local mixture fraction gradient is sufficiently high:  Combustion occurs in a thin layer in the vicinity of this surface • Locally introduce an orthogonal coordinate system x , x , x attached to the 1 2 3 surface of stoichiometric mixture • x points normal to the surface Z , x and x lie within the surface 1 st 2 3 • Replace coordinate x by mixture fraction Z 1 and x , x and t by Z = x , Z = x and t = t 2 3 2 2 3 3 22 Flamelet structure of a diffusion flame • Here temperature T, and similarly mass fractions Y , will be expressed as i function of mixture fraction Z • By definition, the new coordinate Z is locally normal to the surface of stoichiometric mixture • With the transformation rules: we obtain the temperature equation in the form • Transformation of equation for mass fractions is similar 23 Flamelet structure of a diffusion flame • If flamelet is thin in the Z direction, an orderofmagnitude analysis similar to that for a boundary layer shows that is the dominating term of the spatial derivatives • This term must balance the terms on the righthand side • All other terms containing spatial derivatives can be neglected to leading order • This is equivalent to the assumption that the temperature derivatives normal to the flame surface are much larger than those in tangential direction 24 Flamelet structure of a diffusion flame • Time derivative important if very rapid changes occur, e.g. extinction • Formally, this can be shown by introducing the stretched coordinate x and the fast time scale s •ε is small parameter, the inverse of a large Damköhler number or large activation energy, for example, representing the width of the reaction zone 25 Flamelet structure of a diffusion flame • If the time derivative term is retained, the flamelet structure is to leading order described by the onedimensional timedependent flamelet equations • Here is the instantaneous scalar dissipation rate at stoichiometric conditions • Dimension 1/s  Inverse of characteristic diffusion time • Depends on t and Z and acts as a external parameter, representing the flow and the mixture field 26 Flamelet structure of a diffusion flame • As a result of the transformation, the scalar dissipation rate implicitly incorporates the influence of convection and diffusion normal to the surface of the stoichiometric mixture • In the limit c  0, equations for the homogeneous reactor are obtained st 27 Structure of nonpremixed laminar flames Temperature and CH Profiles for Different Scalar Dissipation Rates 2500 2500 a = 0.01/s Methane/Air Diffusion Flame a = 100/s 2000 a = 950/s 2000 c 1500 1500 1000 1000 Propane Counterflow 500 Diffusion Flame 500 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.02 0.01 0 0.01 0.02 0.03 Mixture Fraction y m Temperature K Temperature KSteady solutions of the Flamelet equation: The SShaped Curve • Burning flamelet correspond to the upper branch of the Sshaped curve • If c is increased, the curve is st traversed to the left until c is q reached, beyond which value only the lower, nonreacting branch exists Quenching • Thus at c = c the quenching of st q the diffusion flamelet occurs • The transition from the point Q to the lower state corresponds to the unsteady transition 29 Steady solutions of the Flamelet equation: The SShaped Curve • The neglect of all spatial derivatives tangential to the flame front is formally only valid in the thin reaction zone around Z = Z st • There are, however, a number of typical flow configurations, where is valid in the entire Zspace • As example, the analysis of a planar counterflow diffusion flame is included in the lecture notes 30 LES of Sandia Flame D with Lagrangian Flamelet Model Curvature corresponds to source term Planar Counterflow Diffusion Flame: Analytic Solution • Counterflow diffusion flames • Often used • Represent onedimensional diffusion flame structure • Flame embedded between two potential flows, if • Flow velocities of both streams are sufficiently large and removed from stagnation plane 32 The Planar Counterflow Diffusion Flame Flow equations and boundary conditions • Prescribing the potential flow velocity gradient in the oxidizer stream the velocities and the mixture fraction are there • Equal stagnation point pressure for both streams requires that the velocities in the fuel stream are 33 The Planar Counterflow Diffusion Flame • The equations for continuity, momentum and mixture fraction are given by 34 Example: Analysis of the Counterflow Diffusion Flame • Introducing the similarity transformation one obtains the system of ordinary differential equations in terms of the nondimensional stream function and the normalized tangential velocity 35 Example: Analysis of the Counterflow Diffusion Flame • Furthermore the ChapmanRubesin parameter C and the Schmidt number Sc are defined • The boundary equations are • An integral of the Zequation is obtained as where the integral I(h) is defined as 36 Example: Analysis of the Counterflow Diffusion Flame • For constant properties r = r , C = 1 f = h satisfies  and • The instantaneous scalar dissipation rate is here where and have been used 37 Example: Analysis of the Counterflow Diffusion Flame • When the scalar dissipation rate is evaluated with the assumptions that led to one obtains • For small Z one obtains with l‘ Hospital's rule • Therefore, in terms of the velocity gradient a the scalar dissipation rate becomes 2 showing that c increases as Z for small Z 38 Results of Analysis of the Counterflow Diffusion Flame • Mixture fraction field described as • From this follows scalar dissipation rate as • This provides • Relation between strain rate and scalar dissipation rate • Mixture fraction dependence of scalar dissipation rate, often used in solving flamelet equations 39 Diffusion Flame Structure of MethaneAir Flames • Classical Linan onestep model with a large activation energy is able to predict important features such as extinction, but for small values of Z it st, predicts the leakage of fuel through the reaction zone • However, experiments of methane flames, on the contrary, show leakage of oxygen rather than of fuel through the reaction zone 40 Diffusion Flame Structure of MethaneAir Flames • A numerical calculation with the fourstep reduced mechanism has been performed for the counterflow diffusion flame in the stagnation region of a porous cylinder 41 Diffusion Flame Structure of MethaneAir Flames • Temperature profiles for methaneair flames • Second value of the strain rate corresponds to a condition close to extinction  Temperature in the reaction zone decreases 42 Diffusion Flame Structure of MethaneAir Flames • Fuel and oxygen mass fraction profiles for methaneair flames • The oxygen leakage increases as extinction is approached 43 Diffusion Flame Structure of MethaneAir Flames • An asymptotic analysis by Seshadri (1988) based on the fourstep model shows a close correspondence between the different layers identified in the premixed methane flame and those in the diffusion flame 44 Diffusion Flame Structure of MethaneAir Flames • The outer structure of the diffusion flame is the classical BurkeSchumann structure governed by the overall onestep reaction with the flame sheet positioned at Z = Z st • The inner structure consists of a thin H CO oxidation layer of thickness of order 2 e toward the lean side and a thin inner layer of thickness of order d slightly toward the rich side of Z = Z st • Beyond this layer, the rich side is chemically inert, because all radicals are consumed by the fuel 45 Diffusion Flame Structure of MethaneAir Flames • Results from numerical Simulation of Methane/Air diffusion flame 46 Diffusion Flame Structure of MethaneAir Flames • The comparison of the diffusion flame structure with that of a premixed flame shows that • Rich part of the diffusion flame corresponds to the upstream preheat zone of the premixed flame • Lean part corresponds to the downstream oxidation layer • The maximum temperature corresponds to the inner layer temperature of the asymptotic structure 47 Diffusion Flame Structure of MethaneAir Flames • The plot of the maximum temperature also corresponds to the upper branch of the Sshaped curve • The calculations agree well with numerical and experimental data and they also show the vertical slope of 0 1 T versus c which st corresponds to extinction 48 Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Fundamentals and mass balances of combustion systems • Introduction • Thermodynamics, flame • Counterflow diffusion flame temperature, and equilibrium • Flamelet structure of diffusion flames • Governing equations • FlameMaster flame calculator • Laminar premixed flames: Kinematics and Burning Velocity • Single droplet combustion • Laminar premixed flames: Flame structure • Laminar diffusion flames 49 FlameMaster Flame Calculator • FlameMaster: A C++ Computer Program for 0D Combustion and 1D Laminar Flame Calculations • Premixed and nonpremixed • Steady and unsteady • Emphasis on pre and postprocessing • Sensitivity analysis • Reaction flux analysis • At request, available online at http://www.itv.rwthaachen.de/en/downloads/flamemaster/ FlameMaster Flame Calculator • Example: Shock tube, homogeneous reactor Methylcyclohexane species NOctanol Ignition Delay Times time histories in shock tube FlameMaster Flame Calculator • Example: Flow reactor Methylcyclohexane species time histories in constant pressure plug flow reactor FlameMaster Flame Calculator • Example: Jet stirred reactor NDodecane oxidation in jet stirred reactor FlameMaster Flame Calculator • Example: Reaction flux analysis FlameMaster Flame Calculator • Example: Laminar burning velocities FlameMaster Flame Calculator • Example: Premixed flame structure Methylcyclohexane species profiles in premixed burner stabilized flame FlameMaster Flame Calculator • Example: Flamelet libraries Flamelet library for methane/air nonpremixed combustion 8 8 Methane/Air Diffusion Flamelet Library c Methane/Air Diffusion Flamelet Library 7 7 Variation in Mixture Fraction Variance 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.2 0 0.2 0.4 0.6 0.8 1 1.2 Mixture Fraction Mean Mixture Fraction T/T 2 Normalized Mean TemperatureFlameMaster Flame Calculator • FlameMaster: A C++ Computer Program for 0D Combustion and 1D Laminar Flame Calculations • Premixed and nonpremixed • Steady and unsteady • Emphasis on pre and postprocessing • Sensitivity analysis • Reaction flux analysis • At request, available online at http://www.itv.rwthaachen.de/en/downloads/flamemaster/ Course Overview Part I: Fundamentals and Laminar Flames • Introduction • Fundamentals and mass balances of combustion systems • Introduction • Thermodynamics, flame • Counterflow diffusion flame temperature, and equilibrium • Flamelet structure of diffusion flames • Governing equations • FlameMaster flame calculator • Laminar premixed flames: Kinematics and Burning Velocity • Single droplet combustion • Laminar premixed flames: Flame structure • Laminar diffusion flames 59 Spray Combustion: Gas Turbine Combustion Chamber Quelle: C. Edwards, Stanford University 60 Modelling Multiphase Flows • EulerEuler Approach • EulerLagrange Approach  All phases: Eulerian description  Fluid phase: continuum  NavierStokes Equations  Conservation equation for each phase  Dispersed phase is solved by tracking a large number of  One Phase per Volume particles element → Volume Fraction  The dispersed phase can  Phasephase interaction exchange momentum, mass,  Surfacetracking technique and energy with the fluid applied to a fixed Eulerian phase mesh 61 Dispersed Phase: Droplets • Lagrangian frame of reference • Droplets  Diameter (evaporation)  Temperature (heat transfer)  Deformation (aerodynamic forces)  Collision, breakup, … • Source terms along droplet trajectories • Stochastic approaches:  Monte Carlo method  Stochastic Parcel method 62 Lagrangian Description: Balance equations • Mass balance (single droplet) • Balance of energy (single droplet) • Momentum balance (single droplet) • F : Drag W,i • F : Weight/buoyant force G,i • …: Pressure/virtual/Magnus forces,… 63 Coupling Between the Discrete and Continuous Phases • Mass • Momentum • Energy Coupling Between the Discrete and Continuous Phases  Continuous phase impacts the discrete phase (oneway coupling)  + effect of the discrete phase trajectories on the continuum (source terms, twoway coupling)  + interaction within the discrete phase: particle/particle (fourway coupling) 64 Single Droplet Combustion • Multiphase combustion → phase change during combustion process: Liquid  gas phase • Theoretical description: Single Droplet Combustion • Aim: Mass burnig rate dm/dt as function of  Chemical properties of droplet and surrounding: mixture fraction Z  Thermodynamical properties: Temperature T, density ρ, pressure p  Droplet size and shape: diameter d 65 Single Droplet Combustion • Assumptions  Small droplets which follow the flow very closely  Velocity difference between the droplet and the surrounding fuel is zero  Quiescent surrounding  Spherically symmetric droplet  Neglect buoyant forces  Fuel and oxidizer fully separated → Combustion where the surface of stoichiometric mixture surrounds the single droplet  Diffusion flame  Evaporation and combustion process: quasisteady 66 Single Droplet Combustion Evaporating droplet T u , T , Y 2 2 O,2 κ = const. r u , T , Fuel (Spray) + Inert Gases 1 1 Diffusion flame T Oxidizer κ = const. Fuel+ Inert gases r Flame Burning droplet 67 Single Droplet Combustion • Expected temperature and mixture fraction profiles: 68 Single Droplet Combustion • Quasi stationary evaporation and combustion of a spherically symmetric droplet in Quiescent surrounding  One step reaction with fast chemistry  Le = 1 → Balance equations:  Momentum equation: p = const. 2  Conservation of mass: r ρu = const.  Temperature  Mixture Fraction 69 Single Droplet Combustion • Temperature boundary conditions „+“ Gas phase „“ Liquid phase 70 Single Droplet Combustion • Mixture Fraction boundary conditions „+“ Gas phase „“ Liquid phase 71 Single Droplet Combustion • Temperature BCS:  Enthalpy of evaporation h l  Temperature within the droplet T = const. l  T is boiling temperature T = T (p) L l s • Mixture Fraction BCS:  Difference between the mixture fraction within the droplet and that in the gas phase at the droplet surface 72 Single Droplet Combustion 3 BCS • Quasistationarity: R = const. • BCS in Surrounding: 2 BCS • Integration of the continuity equation leads to Eigenvalue • Mass flux at r equals mass flux at r + dr and at r = R 73 Single Droplet Combustion • Coordinate transformation: • Relation between η und ζ: • Integration and BC ζ = 0 at η = 0 → η = 1 – exp(–ζ) • At r = R → η = 1 – exp(–ζ ) and therefore ζ = – ln(1 – η ) R R R R 74 Single Droplet Combustion • From the equations for temperature and mixture fraction it follows in transformed coordinates: • Transformed BCS • Solution of the mixture fraction 75 Single Droplet Combustion • Temperature solution where Z = h • Known Structure→ Compares to the flamelet equations • We consider the BurkeSchumannsolution  T : Temperature in the surrounding 2  T : Temperature at droplet surface l 76 Single Droplet Combustion • At fuel rich side • Problem:  Temperature T not known 1  Needed to determine T (Z) in the unburnt mixture u 77 Single Droplet Combustion • From BC and it follows • T is a hypothetical temperature 1 corresponding to the fuel if one considers the droplet as a point source of gaseous fuel 78 Single Droplet Combustion • Result: Nondimensional mass burning rate using ζ = – ln(1 – η ): R R • RHS is not a function of the droplet radius • Mass burning rate 79 Single Droplet Combustion • Approximately: ρD ≈ (ρD) ≈ const. → ref → Mass burning rate is proportional to R → Assumptions:  Quasi stationary diffusion flame surrounding the droplet  Constant temperature T within the droplet l 80 Burnout time → It is possible to determine the time needed to burn a droplet with initial radius R • Burnout time: 2 • This is called d law of droplet combustion • It represents a very good first approximation for the droplet combustion time and has often be confirmed by experiments. 81 Single Droplet Evaporation Radius of the surrounding diffusion flame • We want to calculate the radial position of the surrounding flame:  From ρD ≈ (ρD) ≈ const. → ref  With η = 1 – exp(–ζ) and Z = η  1 – Z = exp(–ζ ) → st st → Flame radius → For sufficiently small values of Z the denominator may be approximated st by Z itself showing that ratio r /R may take quite large values. st st 83 Summary Part I: Fundamentals and Laminar Flames • Introduction • Fundamentals and mass balances of combustion systems • Introduction • Thermodynamics, flame • Counterflow diffusion flame temperature, and equilibrium • Flamelet structure of diffusion flames • Governing equations • FlameMaster flame calculator • Laminar premixed flames: Kinematics and Burning Velocity • Single droplet combustion • Laminar premixed flames: Flame structure • Laminar diffusion flames 84
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