Ignition in the laminar boundary layer of a heated plate

Abstract

The present analysis considers ignition and combustion in the laminar boundary layer of a constant temperature, semi-infinite flat plate. A one step unopposed "global" reaction following any order reaction kinetics with temperature dependence according to the Arrhenius rate law is assumed. For the case where the Prandtl and Schmidt numbers are equal, the determination of a similarity function relating the species concentrations to the local temperature greatly simplifies the analysis. The similarity function is shown to be equal to the dimensionless streamwise velocity when the Prandtl and Schmidt numbers are both equal to unity. A general analytic solution for the N'th approximation to the temperature and concentration profiles in the reacting laminar boundary layer is obtained. For all values of plate temperature and free stream velocity, it is found that for some finite distance downstream of the leading edge the plate acts as a heat source; at all points downstream of this characteristic length, however, the plate acts as a heat sink. This characteristic length is closely related to the "flame attachment distance" and is indicative of the minimum plate length required to stabilize a laminar deflagration flame. Although the characteristic length is always finite, it is found that for plate temperatures below a critical threshold band, this length increases so enormously that name attachment cannot occur on physical apparatus of reasonable finite dimension. Inasmuch as the classical boundary layer assumptions are invalidated in the immediate region of flame attachment, the complete development of the laminar flame front cannot be obtained within the framework of the present boundary layer type analysis.