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III-9

In Equation (13) łt is possible for the B term to be negative and larger in magnitude than the A term. In such cases, the theory predicts that turbulence decreases the flame speed. However, it seems from estimates of the terms in Eąuation (13) that this situation is unlikely to occur in real flames. Nevertheless, it cannot be ruled out.

To cóntrast the new results with those of earlier work, notę that the way in which factors 1/6 and 0 appear in Eąuation (13) shows that an increase in turbulence intensity, an increase in eddy size, or an increase in activation energy will increase the departure of the flame speed from its laminar value. If the skewness term is smali compared with the variance term, then Eąuation (13) agrees with an earlier result⁸ conceming the dependence of flame speed on intensity (v_T — v_L proportional to e ), although other models¹ usually have given a somewhat weaker dependence on e . Earlier work has not predicted the dependence on /, sińce the two-zone structure needed to achieve amplification of fluctuations to a level proportional to l was not included. Also, there are no earlier theoretical results conceming the dependence of v_T on the activation energy.

3.5 Comparison of Theoretical and Experimental Turbulent Flame Speeds

There are many experimental results on the dependence of turbulent flame speeds on turbulence intensity¹, but few for intensities Iow enough for the preceding theoretical considerations to be applicable. The relationship v_T ~ often is observed for large e .

There is little data on the dependence of flame speed on turbulence scalę. In early work, some authors have reported that v_T is independent of /, while others have reported that v_T increases with /¹ . This may be consistent with the fact that the preceding model predicts an increase with / only if 1/6    1 ; otherwise (//5)² A

and (//5)³B are independent of 1/6 , and there is predicted to be no dependence on /. Recently, a relatively detailed experimental study of the dependence of v_T on e and / has been completed for propane-air flames⁹. Results are shown in Figurę 4. It is seen that the scalę dependence is distinctly different at high and Iow intensities The strong increase in flame speed with increasing scalę at the lower intensities is qualitatively consistent with the prediction of the theory.

4. LAMINAR DIFFUSION FLAME IN A SHEAR FLOW

The type of analysis reviewed in Sections 2.3 and 2.4 also can be completed for diffusion flames. To proceed, it is necessary to understand in advance the structure of a one-dimensional, laminar, diffusion flame in the absence of shear. The processes of diffusion and reaction are basically the same as those discussed for the premixed flame. The differences in structure stem from the boundary conditions, fuel and oxidizer initially being separated for the diffusion flame. Usually both fuel and oxidizer are cold, and the temperaturę peaks at a thin reaction zonę and decreases in the broad diffusion zones which exist on each side of the reaction zonę. Unlike the premixed flame, the diffusion flame is not influenced by the ratę of the Chemical reaction, except in the narrow reaction zonę, which often can be approximated well as sheet of negligible thickness. Thus, an expression likc Eąuation (1) describes diffusion-flame structure, but in a First approximation the ratę term may be removed and replaced by suitable continuity conditions for coupling functions. Morę details may be found in References 1 and 3.

An important result of the differing boundary conditions for premixed and diffusion flames is that in the strictly one-dimensional case in infinite space the latter possesses no steady-state solution; diffusion flames evolve transiently. The reason is that in the absence of shear or boundaries, convective-diffusive balances cannot be established on both sides of the reaction zonę. Thus, for 7 = 0 the diffusion flame is unsteady.

For 7 =£ 0 Eąuation (5) again describes the flame structure. The reasoning of Section 2.4 now shows that in the presence of positive flame stretch, convective-diffusive balances can be established on each side of the reaction zonę in the diffusion flame. In view of the existence of the counterflow diffusion flame, physically this must be true. The situation, summarized in Table 3, bears a close correspondence to that for the premixed flame, given in Table 2. Stretched diffusion flames are steep, thin and steady, compressed ones thick and unsteady.

The inflow into a stretched diffusion flame increases the reactant flux to the reaction zonę. This in tum decreases the residence time in the reaction zonę. It is known³ that if the ratio of the residence time to the Chemical reaction time , i.e., Damkóhler*s First similarity parameter D, , becomes too smali, then extinction occurs due to insufflcient time for heat release. Thus, too much stretch will extinguish a diffusion flame; compression will not.

There is a critical Damkóhler number, D, _ext , defining extinction conditions. In “flame-strength” experiments,

7 is gradually increased until D, reaches D_lext . Applying these results to turbulent flows, one sees immediately that, for turbulence scales large compared with diffusive-zone thicknesses for the flame, if the strain ratę y becomes too large, then local extinctions of sheared, laminar, diffusion flames will occur in the turbulent diffusion-flame brush. For sufficiently smali values of 7 , a turbulent diffusion flame of large scalę can be viewed as a collection of sheared laminar diffusion flames, but for sufficiently large values of 7 , it cannot.

It will be seen in the following section that if the turbulent flame is an ensemble of laminar diffusion flames, then some progress can be madę in theoretical analysis of it. For this reason it is important to know when the