# Turbulent Non-Premixed Combustion

###### Turbulent Non-Premixed Combustion
Turbulent NonPremixed Combustion 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 II: Turbulent Combustion • Turbulence • Turbulent Premixed Combustion • Laminar Jet Diffusion Flames • Turbulent NonPremixed • Turbulent Jet Diffusion Flames Combustion • Modelling Turbulent Combustion • Applications 2 Laminar Jet Diffusion Flames flame length L fuel air 3 Round Laminar Diffusion Flame • Fuel enters into the combustion chamber as a round jet • Forming mixture is ignited • Example: Flame of a gas lighter  Only stable if dimensions are small  Dimensions too large: flickering due to influence of gravity  Increasing the jet momentum → Reduction of the relative importance of gravity (buoyancy) in favor of momentum forces  At high velocities, hydrodynamic instabilities gain increasing importance: laminarturbulent transition 4 Laminar Diffusion Fame: Influence of Gravity 1g 0g 5 Round Laminar Diffusion Flame • Starting point: Conservation equations for stationary axisymmetric boundary layer flow without buoyancy • Continuity: flame length L • Momentum equation in zdirection • Mixture fraction fuel air 6 Round Laminar Diffusion Flame • Schmidt number Sc = μ/ρD • Farfield area  r → ∞: u = u = 0 z r  From zmomentum equation  dp/dz = 0 • Boundary layer flow: 0 flame length L • Incompressible round jet  Quiescent ambient  Constant density  No buoyancy → Similarity solution • Simularity coordinate η = r/z (Schlichting, „Boundary Layer Theory“) fuel air 7 Round Laminar Diffusion Flame • If density not constant → Transformation flame length L • a: Distance of the virtual origin of the jet from the nozzle exit • For ρ = const. und a → 0 • Implies linear spreading of the roung jet fuel air 8 Round Laminar Diffusion Flame • Introduction of a stream function → Continuity equation identically satisfied • Applying the transformation rules to the convective terms in the momentum and mixture fraction equations yields 9 Round Laminar Diffusion Flame • Such manipulations are always possible for twodimensional stationary boundary layer flows, if a stream function and a similarity coordinate ζ ≠ f(r) can be introduced • The diffusive terms become • C: ChapmanRubesinParameter • For constant density (with η = r/ζ and μ = μ ): C = 1 ∞ 10 Round Laminar Diffusion Flame • Formal transformation of the momentum and concentration equations and assumption that C = f(ζ,η) • Ansatz for stream function and for the velocities • u und u can be expressed as a function of the nondimensional stream function F and its z r derivatives 11 Round Laminar Diffusion Flame • From the momentum equation  • Similarity solution only exists, if F ≠ f(ζ) • Then, u is proportional to 1/ζ (see previous slide) z → velocity decreases linearly with 1/(z + a) • Prerequesites: Boundary conditions and C are independent of z (e. g. u = 0 and u = 0 for η → 0) z r 12 Round Laminar Diffusion Flame • Equation for the nondimensional stream function • Let ω = Z(z,r)/Z (z), ratio of the mixture fraction Z (z) to its value at r = 0 a a • Applying the same transformations to the ωequation yields • In case of a similarity solution 13 Round Laminar Diffusion Flame • If C = const.: • The assumption C = const. Holds if and ρμ/ρ μ = const. m ∞ • C = const. Often not a good assumption, since 14 Round Laminar Diffusion Flame • Constant of integration γ can be determined from the condition that the jet momentum is independent of ζ • Substitution of the solution into the momentum balance yields • ρ : density of the fuel stream 0 • Reynolds number Re = u d/ν z,0 ∞ 15 Round Laminar Diffusion Flame • Analogously for the mixture fraction (with Z = 1) 0 → Mixture fraction on the centerline Z (z) = Z(z,r=0): a → Z decreases with 1/ζ (as the velocity) a 16 Round Laminar Diffusion Flame • Determination of the flame contour r as function of z from the condition • Flame contour intersects centerline, flame r = 0, if Z = Z length L a st • Corresponding value of z defines the flame length • Valid for laminar jet flames without buoyancy fuel air 17 Round Laminar Diffusion Flame • For a given nozzle diameter, L increases linearly with the Reynolds number Re laminar fully developed flame transition turbulent flame flame length Sc =0,72 t L Reynolds number Re fuel air 18 flame length L/d Course Overview Part II: Turbulent Combustion • Turbulence • Turbulent Premixed Combustion • Laminar Jet Diffusion Flames • Turbulent NonPremixed • Turbulent Jet Diffusion Flames Combustion • Modelling Turbulent Combustion • Applications 19 Turbulent Jet Diffusion Flame • Shear flow at nozzle exit • Flow instabilities (KelvinHelmholtzinstabilities) → laminarturbulent transition • Ring shaped turbulent shear layer propagates in radial direction • Merging after 10 to 15 nozzle diameters downstream • Streamlines are parallel in th potential core • Velocity profile reaches self similar state after 2030 nozzle diameters 20 Round Turbulent Diffusion Flame • Linear reduction of velocity along central axis • Linear increase of jet width • Assumption: fast chemical reaction → Scalar quantities such as temperature, concentration and density as function of mixture fraction Z • Turbulent flow with variable density → Favreaveraged boundary layer equations 21 Linear Propagation of (turbulent) Jet 22 Round Turbulent Diffusion Flame • Assumptions:  Axisymmetric jet flame  Neglecting buoyancy  Neglecting molecular transport as compared to turbulent transport  Turbulent transport modeled by Gradient Transport model  Sc = ν /D t t t • Using Favre averaging and the the boundary layer assumption we obtain a system of twodimensional axisymmetric equations 23 Round Turbulent Diffusion Flame • Continuity equation • Momentum equation in zdirection • Mean mixture fraction 24 Round Turbulent Diffusion Flame • Requires solving of equations for k and ε to determine ν t • Round turbulent jet: ν approximately constant t • Analogous for round laminar jet: Laminar Turbulent 25 Round Turbulent Diffusion Flame • Special case: Jet in quiescent ambient  Treatment of turbulent equations like those in a laminar round jet case  Using the laminar theory • Similarity coordinate Laminar Turbulent • ChapmanRubesinParameter 26 Round Turbulent Diffusion Flame • Turbulent ChapmanRubesinParameter approximately constant→ • Integration constant γ, containing fuel density and reference viscosity • The Favreaveraged velocity decreases proportional to 1/ζ = 1/(z + a), just like in the laminar case 27 Round Turbulent Diffusion Flame • Mean mixture fraction with → Mixture fraction decreases proportional to 1/(z + a) on the jet axis → Progression of profiles along jet axis resembles those of the laminar case • Also applies to the contour of the stoichiometric mixture 28 Round Turbulent Diffusion Flame • Flame length L of round turbulent diffusion flame: Distance z from the nozzle, where the mean mixture fraction on the axis equals Z st • Comparison with experimental correlations (Hawthorne, Weddel and Hottel (1949))  With u d/ν = 70 and Sc =0,72 z,0 t,ref t 1/2  Complete agreement for C = (ρ ρ ) /ρ 0 st ∞ 29 Round Turbulent Diffusion Flame const. linear ≈ 70 laminar fully developed flame transition turbulent flame Reynolds number Re 30 flame length L/d Experimental Data: Round Turbulent Diffusion Flame • Comparison of experimental results and simulations with chemical equilibrium • Concentration of radicals and emissions cannot be described by infinitely fast chemistry 31 Summary Part II: Turbulent Combustion • Turbulence • Turbulent Premixed Combustion • Laminar Jet Diffusion Flames • Turbulent NonPremixed • Turbulent Jet Diffusion Flames Combustion • Modelling Turbulent Combustion • Applications 32
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