The influence of a flameholder on a plane flame, including its static stability

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Abstract
The plane pre-mixed flame adjacent to a coplanar porous-plug type of burner is analysed with the aid of large-activation energy asymptotics. It is shown th at there is a unique relation between the mass flux of reactants through the holder and the burnt-gas temperature for a given value of the mixture equivalence ratio. A simple linear heat-flux-temperature model of the flameholder introduces a constant of proportionality called the conductance of the holder. The conductance has a strong influence on the details of flame behaviour; in particular the heat flux from the preheat regions of the flame into the flameholder is governed by the conductance, as is the stand-off distance between the face of the holder and the flame-sheet. It is also shown that, despite the foregoing facts, the heat flux that must be provided at the holder to or from some outside agency, such as a coolant, is actually independent of the conductance. A detailed and general analysis of the flame-flameholder system identifies a critical value for the conductance (that depends upon the mixture equivalence ratio). In addition a dimensionless number, called a, that is equal to the specific heat multiplied by the mass flux of the reactants divided by the conductance, has an important bearing on the type of equilibrium that exists between the flame and the flameholder. If a < 1 all equilibrium solutions have a mass flux through the holder that is less than the adiabatic mass flux at the given equivalence ratio; they require that heat shall be abstracted from the system by the coolant. When a > 1 the equilibrium mass flux must exceed the adiabatic value, and the ‘coolant’ must supply heat to the system. A plot of inlet flow speed against equivalence ratio for several fixed values of the heat-loss to the coolant per unit area per unit time shows that there are either two equilibrium inlet speeds for a given equivalence ratio or none at all. This result is in in accord with the experimental observations of Botha & Spalding (*954)- The existence of a distinguishable spatial origin (the flameholder surface) means that the static stability of the flame, considered as a single plane sheet, can be defined. Domains of static stability and instability are identified, and help to explain observations of flame behaviour on a coplanar holder. Static stability or instability complements diffusional instabilities; however, static stability boundaries are not affected by the Lewis number, as are those of the diffusional mechanism. Despite the fact that most of the flames in streams the mass flux of which exceeds the adiabatic value at a given equivalence ratio are statically unstable, and are therefore not to be found in practice, the analysis predicts th at plane flames can be stabilized on a holder in some circumstances at flow rates greater than the adiabatic if the flow speed exceeds a calculable minimum.