Turbulent Non-Premixed Combustion · • For ρ = const. und a → 0 • Implies linear spreading...

Turbulent Non-Premixed Combustion

Combustion Summer School

Prof. Dr.-Ing. Heinz Pitsch

2018

Course Overview

2

• Turbulence

• Turbulent Premixed Combustion

• Turbulent Non-Premixed Combustion

• Turbulent Combustion Modeling

• Applications

• Laminar Jet Diffusion Flames • Turbulent Jet Diffusion Flames

Part II: Turbulent Combustion

flame length L

fuel air

Laminar Jet Diffusion Flames

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Laminar Jet 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: laminar-turbulent transition

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Laminar Diffusion Fame: Influence of Gravity

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1g 0g

Laminar Jet Diffusion Flame (Governing Equations)

• Starting point: Conservation equations for stationary axisymmetric boundary layer flow without buoyancy

• Continuity:

• Momentum equation in z-direction

• Mixture fraction

6

flame length L

fuel air

Laminar Jet Diffusion Flame (Assumptions + BC)

• Schmidt number Sc = μ/ρD • Farfield area

− r → ∞: uz = ur = 0 − From z-momentum equation dp/dz = 0

• Boundary layer flow:

• Incompressible round jet − Quiescent ambient − Constant density − No buoyancy → Similarity solution

• Simularity coordinate η = r/z (Schlichting, „Boundary Layer Theory“)

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0 flame length L

fuel air

Laminar Jet Diffusion Flame (Similarity Coordinates)

• If density not constant → Transformation

• 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

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flame length L

fuel air

Laminar Jet Diffusion Flame (Stream Function)

• 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

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Laminar Jet Diffusion Flame (Transformation Rules)

• Applying the transformation rules to the convective terms in the momentum and mixture fraction equations yields

• The diffusive terms become

• C: Chapman-Rubesin-Parameter

• For constant density (with η = r/ζ and μ = μ∞): C = 1 10

Laminar Jet Diffusion Flame (Non-dim. Stream Func.)

• Formal transformation of the momentum and concentration equations and assumption that C = f(ζ,η)

• With ansatz for non-dimensional stream function F for the velocities follows

• uz und ur can be expressed as a function of the nondimensional stream function F and its derivatives

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Laminar Jet Diffusion Flame (Transformation)

• From the momentum equation

• Similarity solution only exists, if F ≠ f(ζ) • Then, uz is proportional to 1/ζ (see previous slide)

→ velocity decreases linearly with 1/(z + a) • Prerequesites: Boundary conditions and C are independent of z

(e. g. uz = 0 and ur = 0 for η → 0)

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Laminar Jet Diffusion Flame (Resulting Equations)

• Equation for the nondimensional stream function

• Let ω = Z(z,r)/Za(z), ratio of the mixture fraction Za(z) to its value at r = 0 • Applying the same transformations to the ω-equation yields

• In case of a similarity solution

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Laminar Jet Diffusion Flame (Analytic Solution)

• Integration for C = const. yields:

where γ is integration constant

• The assumption C = const. Holds if and ρμ/ρm μ∞ = const.

• C = const. Often not a good assumption, since

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Laminar Jet Diffusion Flame (Integration Constant γ)

• 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

• ρ0: density of the fuel stream • Reynolds number Re = uz,0d/ν∞

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Laminar Jet Diffusion Flame (Centerline Mixt. Fraction)

• Analogously for the mixture fraction (with Z0 = 1)

→ Mixture fraction on the centerline Za(z) = Z(z,r=0):

→ Za decreases with 1/ζ (as the velocity)

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flame length L

fuel air

Laminar Jet Diffusion Flame (Flame Length)

• Determination of the flame contour r as function of z from the condition

• Flame contour intersects centerline, r = 0, if Za = Zst

• Corresponding value of z defines the flame length

• Valid for laminar jet flames without buoyancy

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Laminar Jet Diffusion Flame

• For a given nozzle diameter, L increases linearly with the Reynolds number Re

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flame length L

fuel air

Reynolds number Re

transition fully developed turbulent flame

laminar flame

flam

e le

ngth

L/d

Sct=0,72

Course Overview

19

• Turbulence




• Applications



Turbulent Jet Diffusion Flame

• Shear flow at nozzle exit • Flow instabilities

(Kelvin-Helmholtz-instabilities) → laminar-turbulent transition

• Ring shaped turbulent shear layer propagates in radial direction

• Merging after 10 to 15 nozzle diameters downstream

• Streamlines are parallel in potential core • Velocity profile reaches self similar

state after 20-30 nozzle diameters

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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 → Favre-averaged boundary layer equations

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Linear Propagation of (turbulent) Jet

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• Assumptions: − Axisymmetric jet flame − Neglecting buoyancy − Neglecting molecular transport as compared

to turbulent transport − Turbulent transport modeled by

Gradient Transport model − Sct = νt/Dt

• Using Favre averaging and the the boundary layer assumption we obtain a system of two-dimensional axisymmetric equations

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• Continuity equation

• Momentum equation in z-direction

• Mean mixture fraction

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• Requires solving of equations for k and ε to determine νt • Round turbulent jet: νt approximately constant • Analogous for round laminar jet:

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Turbulent Laminar


• Special case: Jet in quiescent ambient − Treatment of turbulent equations like those in a laminar round jet case − Using the laminar theory

• Similarity coordinate

• Chapman-Rubesin-Parameter

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Turbulent Laminar


• Turbulent Chapman-Rubesin-Parameter approximately constant→

• Integration constant γ, containing fuel density and reference viscosity

• The Favre-averaged velocity decreases proportional to 1/ζ = 1/(z + a), just like in the laminar case

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• 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

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• Flame length L of round turbulent diffusion flame: Distance z from the nozzle, where the mean mixture fraction on the axis equals Zst

• Comparison with experimental correlations (Hawthorne, Weddel and Hottel (1949)) − With uz,0d/νt,ref = 70 and Sct=0,72 − Complete agreement for C = (ρ0 ρst)1/2/ρ∞

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30

const.

≈ 70

linear

Reynolds number Re

transition fully developed turbulent flame

laminar flame

flam

e le

ngth

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

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Summary

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• Turbulence




• Applications



Turbulent Non-Premixed Combustion · • For ρ = const. und a → 0 • Implies linear spreading...

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