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# Relationship between flame height and fuel flow rate

### Laminar diffusion flames: L \sim \dot{m}_{fuel}

Hottel and Hawthorne (1949) suggested that a hydrogen-fuelled flame would undergo transition from laminar to turbulent flow at a Reynolds number of around 2000.

Flame length is defined as the distance between the nozzle exit and the point on the jet centreline whe-re the mean mixture fraction equals the stoichiometric mixture fraction.

### Axisymmetric jet ejecting vertically upward –

\frac{L + x_0} {d} = \frac {5.3} {Z_{st}} \left( \frac{\rho_0}{\rho_{st}} \right)^{1/2},

where x_0 is the virtual origin, d is the jet diameter, Z_{st} is the stoichiometric mixture fraction, \rho_0 is the ambient density and \rho_{st} is the density of a stoichiometric mixture.

Peters and Göttgens (1991) presented a relation between a non-dimensionless flame length and the Froude number for buoyant diffusion flames. Peters (2000) included a graph showing the flame length against Froude number for propane-air diffusion flames.

### Vertical jet ejecting upwards – quiescent ambient conditions

The flame will be longer if the jet is ejecting vertically upwards and buoyancy is taken into account.

### Vertical jet ejecting upwards – crosswind ambient conditions

The momentum of the crosswind will cause the flame to bend over with the wind. Even a crosswind with relatively low momentum would make a high momentum jet bend over. Muppidi and Mahesh (2005) used DNS to simulate high momentum jets in a low momentum cross flow; A jet with a mo-mentum ratio of 1.5, Mjet/Mcrossflow, would be deflected at y/d \lambda 1.5 and a jet with a momentum ratio of 5.7 would be deflected at y/d \lambda 6. The jet will exhibit the characteristic kidney-shape due to counter-rotating vortices, which draws ambient air into the jet. The mechanism responsible for the formation of this vortical structure appears to be relatively insensitive to Reynolds number of the jet provided that the jet momentum is high enough for the jet to penetrate the cross-flow. It is conjectured that the breakdown of the structure shortly downstream of the bending of the jet is due to interaction between the vortices and Kelvin-Helmholtz like instabilities. Horseshoe vortices are generated near the ground as the cross-flow moves around the jet.

### Vertical jet ejecting downwards – not interacting with the ground

A jet flame directed vertically downward will be shorter than if it was ejecting upwards, if buoyancy is taken into account and the jet does not interact with the ground.

### Horizontal free jet

Buoyancy forces eventually exceed the momentum forces of the jet, and the flame tip rises. Co-flowing wind would make the jet longer, delaying the rise of the flame tip due to buoyancy.

### Horizontal jet near ground

There will be jet-ground interaction which will affect the flame length. Co-flowing wind would make the jet longer, while an opposing wind would reduce the flame length

### Lifted flames

Jet flames with relatively low exit velocity will be stabilised on or attach to the nozzle exit. The diffu-sion flame sheet will become stretched and finally distorted if a critical exit velocity is exceeded and the flame will be stabilised some distance, the lift-off height, downstream of the nozzle exit. The lift-off height increases with increasing jet exit velocity, until a critical velocity has been reached at which point the flame is blown out. A lifted flame will re-attach to the nozzle if the exit velocity is reduced, but the re-attachment velocity will almost certainly be different from the lift-off velocity due to the hysteresis effect.

Four different mechanisms have been proposed as explanation of the lift-off phenomenon:

• A diffusion flame will be lifted if the velocity gradient at the burner rim exceeds a critical value and will be stabilised at the point where the burning velocity equals the mean flow velocity, Wohl et al. (1949);
• The fuel and air are premixed at the base and the flame will stabilise at the point when the mean flow velocity at the mean stoichiometric mixture contour equals the burning velocity, Vanquickenborn and van Tiggelen (1966). Experiments performed by Eickhoff et al. (1984) showed large entrainment of air at the base, generating premixed conditions at the flame base;
• Peters and Williams (1983) proposed that diffusion flamelet extinction is responsible for the flame stabilisation; and
• Pitts (1989) suggested that cold, isothermal mixing of non-burning fuel jet and air could provide the mechanism by which the flame is stabilised.

Flame quenching is the probable mechanism responsible for lift-off of an initially attached flame, Peters (2000).

Kalghatgi (1981) proposed a correlation to calculate the flame lift-off height based on maximum lami-nar flame speed. Kalghatgi (1981) also proposed a correlation to estimate the blowout velocity for hydrocarbon fuels for burners with different diameters:

U_b = C_b S^2_{b,max} r_{jet},

where C_b is an experimental constant, S_{b,max} is the maximum laminar flame speed and r_{jet} is the radius of the burner.

Eickhoff, H., Lenze, B., and Leuckel, W., Experimental investigation on the stabilization mechanism of jet diffusion flames, Twentieth Symposium (International) on Combustion, The Combustion Insti-tute, Pittsburgh, Pennsylvania, USA, 1984.
Hottel, H. C., and Hawthorne, W. R., Diffusion in laminar flame jets, 3rd Symposium on Combustion, Flames and Explosions, pp. 254-266, 1949.
Kalghatgi, G. T., Blow-out stability of gaseous jet diffusion flames. Part I: In still air, Combustion and Flame 26:233-239, 1981.
Muppidi, S., and Mahesh, K., Study of trajectories of jets in crossflow with direct numerical simula-tions, Journal of Fluid Mechanics 530:81-100, 2005.
Peters, N., Turbulent Combustion, Cambridge University Press, Cambridge, United Kingdom, 2000.
Peters, N., and Göttgens, J., Scaling of buoyant turbulent diffusion flames, Combustion and Flame 85:206-244, 1991.
Peters, N., and Williams, F. A., Lift-off characteristics of turbulent jet diffusion flames, AIAA Journal 21(3):423-429, 1983.
Pitts, W. M., Importance of isothermal mixing processes on the understanding of lift-off and blowout of turbulent jet diffusion flames, Combustion and Flame 76:197-212, 1989.
Vanquickenborn, L., and Tiggelen, A. van, The stabilization mechanism of lifted diffusion flames, Combustion and Flame 10:59-69, 1966.
Wee, D. H., Lagrangian Simulation of a Jet in Crossflow at a Finite Reynolds Number, http://web.mit.edu/dhwee/www/lagsimjicf_re/, 2004.