Simplified Descriptions of Laminar Diffusion Flames Burke-Schuman simplified description

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Lecture 21: Simplified Descriptions (of Laminar Diffusion Flames), Soot and Counter-flow Geometry. Simplified Descriptions of Laminar Diffusion Flames Burke-Schuman simplified description Roper solution Constant density solution Variable density approximate solution by Fay

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Lecture 21: Simplified Descriptions (of Laminar Diffusion Flames), Soot and Counter-flow Geometry

• Simplified Descriptions of Laminar Diffusion Flames
• Burke-Schuman simplified description
• Roper solution
• Constant density solution
• Variable density approximate solution by Fay
• Numerical solutions
• Circular, Square and Slot Burners
• Soot Formation and Destruction
• Counterflow Flames

• Assume:

1. Laminar, steady, axisymmetric flow, vertical flame axis, axial diffusion is neglected

2. Equal diffusivity, unity Lewis number, conserved scalar approximation

4. Pressure gradient assumed to be hydrostatic

Conservation Equations: Cylindrical Coordinates, Thin Flame

Conservation of Mass

Conservation of Axial Momentum

Conservation of Species Mass Fractions

Conservation of Energy

Constitutive Relationships: Ideal Gas Law, Lewis Number etc.

Conserved Scalar Equations for Laminar Jet Flame

• Boundary Conditions

At the jet exit plane

Count Unknowns:

Conservation Equations: Laminar Jet Diffusion Flame

Count Unknowns:

Count Equations: Conservation of mass, conservation of axial momentum, conservation of radial momentum, conservation of energy, conservation of N species, Ideal Gas Law, Definition of f, Definition of h, and properties.

Conservation of radial momentum: We did not write and the

text book did not write either. Please write as an exercise!

Examine the term:

Where is the pressure gradient term?

What is pressure gradient equal to?

Non-dimensional Laminar Jet Diffusion Flame

• A dimensionless enthalpy is defined:
• The non-dimensional conservation equations and boundary conditions for h* and f are identical, and therefore h* = f.
• The non-dimensional conservation equations and boundary conditions for h* and f are identical, and therefore h* = f.

Description of Laminar Flame State-relationships

• Laminar Flame State-Relationships imply that all species concentrations are solely functions of the mixture fraction.
• These functions can be determined from experiments involving careful laminar flame measurements that are yield plots of species mass fractions or species mole fractions as functions of mixture fraction.
• Laminar flamelet state relationships do not assume that the chemistry is fast. All that they assume is that the reaction rates are known and are functions solely of mixture fraction.
• Once the reaction rates are defined solely as a function of mixture fraction, the capability to have transient processes is lost.
• This capability can be partially restored by defining the transient combustion processes to occur between one state relationship to the other.

Conservation Equations: Burke-Schumann

Conservation of Species Mass Fractions

Solution involves Bessel Function see 9.55

Conservation Equations: Roper Solution

See equations 9.59, 9.60

Conservation Equations: Fay Solution

See equations 9.57, 9.58 and Table 9.2

• See equations 9.61- 9.70
• Buoyancy versus momentum control
• Jet exit momentum is an important quantity
• Froude number defines the effect of buoyancy on flame length
• Error function and inverse error function of a stoichiometric ratio parameter determine the flame length
• See Figure 9.10 for fuel effects
• In all cases the flame length depends on the volumetric flow rate. A smoker with a mustache and then a cigarette lighter company learned this lesson the hard way. I don’t know who won!!