Simplified Descriptions of Laminar Diffusion Flames Burke-Schuman simplified description

1. 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

2. Laminar Jet Diffusion Flames (Non-premixedJet Flames)

3. Simplified Theoretical Description of Laminar Jet Diffusion Flame • Assume: 1. Laminar, steady, axisymmetric flow, vertical flame axis, axial diffusion is neglected 2. Equal diffusivity, unity Lewis number, conserved scalar approximation 3. Radiation heat transfer treated using radiation heat loss fraction 4. Pressure gradient assumed to be hydrostatic

4. 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.

5. Conserved Scalar Equations for Laminar Jet Flame • Boundary Conditions At the jet exit plane Count Unknowns:

6. 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?

7. 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.

8. 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.

9. 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

10. Conservation Equations: Roper Solution for Non-circular Burners • 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!!