A review of the fluid dynamic problem posed by the laminar jet diffusion flame

Abstract A theory is presented in which the flame length, L∗, of a planar laminar jet diffusion flame is separated into two parts: L∗=li∗+ls∗, where li∗ is the length of the part of the flame in the initial region immediately downstream of the burner exit where the boundary layer similar solutions are not applicable and ls∗ is the length of the part of the flame in the far downstream region where the boundary layer similar solutions are applicable. Boundary layer similar solutions which encompass a detailed consideration of the distributions of the relevant physical quantities are developed. A comparison of experimental data with the theoretical formula for ls∗ indicates that the similar solution does not become valid until the amount of fuel unreacted has dropped to, say, less than ten per cent of the original fuel concentration and that this value decreases rapidly with increasing burner diameter and initial Reynolds number. Further, l s ∗ L ∗ may be ⪡1. The theoretical form of ls∗ indicates that a unit length of flame front in the far downstream flow field where the similar solution applies is a much less efficient combustor than a unit length of flame front near the burner exit where the distributions are definitely non-similar. Thus, for an actual flame, almost all of the combustion takes place along that part of the flame front which lies in the initial non-similar region of the flow field, and a very small part of the combustion takes place along that part of the flame front which lies in the region where the Prandtl boundary layer approximations and the resulting similar solutions are valid. Hence, it is concluded that the similar solution may not be used to estimate, even roughly, the overall characteristics of the flame. Several important conclusions are obtained from a dimensional approach. It is shown that the size of L∗ is governed by the form of the velocity decay law and that the theoretical dependence of L∗ on Reynolds number is governed by the characteristic length associated with the velocity decay law. Following Kaplun's work on the role of the coordinate systems in boundary layer theory, it is shown that within the formal context of the Prandtl boundary layer approximations, the theoretical result for L ∗ w ∗ must always be proportional to the first power of the Reynolds number, in contradistinction to the experimental results. Finally, it is concluded that any analysis of the laminar jet diffusion flame with a view to predicting adequately either the overall characteristics of the flame or the detailed distributions of the physical properties in the flow field must be based upon a method of solution which accounts for (i) the velocity and the temperature distributions on the transverse plane at the burner exit; (ii) the velocity decay law in the immediate downstream region; (iii) the boundary layer region next to the burner lip; and (iv) the heat transfer from the flame region to the solid burner lip. This solution in the initial region will definitely be non-similar. The present work indicates approximately how far downstream this non-similar solution must be carried out before it can be replaced safely by the similar solution; this undoubtedly depends on the individual problem. The analytical determination of the truly laminar jet diffusion flame appears to be much more complicated than was previously recognized; this statement can be made without even considering the possibility of encountering small scale turbulence in the mixing zone of actual flames.