The asymptotic structure of counterflow diffusion flames for large activation energies

TL;DR: In this paper, the structure of steady state diffusion flames is investigated by analyzing the mixing and chemical reaction of two opposed jets of fuel and oxidizer as a particular example, and an Arrhenius one-step irreversible reaction in the realistic limit of large activation energies.

About: This article is published in Acta Astronautica.The article was published on 1974-07-01 and is currently open access. It has received 792 citations till now. The article focuses on the topics: Premixed flame & Diffusion (business).

Summary

1. Introduction

• Two basic classes of non premixed combustion problems can be defined as "initial-value problems", described either by ordinary differential equations with boundary conditions at only one point or by parabolic partial differential equations, and "boundary-value problems", described by ordinary differential equations with two boundary points or by elliptic partial differential equations.
• In Section 7 the authors summarize the results and compare with other numerical results.
• Initial value problems in combustion have been analyzed for large activation energies by Lifian and Crespo [25] .

2. Formulation

• The authors shall analyze the simultaneous mixing and chemical reaction at the stagnation point of two counter-flow streams of fuel and oxidizer.
• A one-step irreversible Arrhenius reaction will be considered, which is first order with respect to both fuel and oxidizer.
• The authors shall use Fick's law, with equal diffusivities of mass and heat, to calculate the diffusion velocities.
• The four different possible regimes that the authors shall encounter for large T a show the existence within the flow field of two regions, where the temperature is given in the first approximation by Eqs. ( 8) to (10) , separated by a thin reaction zone.
• Too to the adiabatic flame temperature the authors may encounter a nearly frozen ignition regime, a partial burning regime, a pre-mixed flame regime and finally a diffusion controlled or diffusion flame regime.

2. Partial burning regime.

• Infinitely thin in the limit T a ~* <», reaction region which acts as sink for the reactants; because the reaction rate is not sufficiently fast, leakage of both reactants through the flame occurs.
• The asymptotic structure of counterfiow diffusion flames 1013.
• The structure of the reaction zone is the same as the structure of the reaction zone of a premixed flame with heat loss toward the equilibrium side of the flame.
• Two possibilities arise depending on the value of a+2j3.

4. Diffusion flame or near-equilibrium regime. For sufficiently high

• Damkohler numbers a diffusion controlled regime exists in which equilibrium is reached, to the first approximation, on both sides of a thin reaction zone.
• The authors shall see below, when analyzing the structure of the thin reaction zone, that for /3 < 1 and sufficiently high activation energies multiple solutions may exist for Damkohler numbers above a minimum "extinction D" and no solution for D below this value.
• There are cases in which the extinction conditions do not occur under this regime but under the previous premixed flame regime.
• The relation between flame temperature and Damkohler number will be obtained, sequentially for the different regimes in the following sections, from the matching conditions between the asymptotic expansions for the temperature in the reaction zone and in the outer regions.

3. Nearly frozen, ignition, regime

• As indicated before, in this regime the Damkohler number is such that the temperature rise, above the frozen flow value, due to the chemical production term, is of order TJlT a , which is small compared with TV, but large enough so as to force us to retain in the equations the nonlinear effects associated with the Arrhenius exponent.
• The authors look for the Damkohler number D that produces, at x -1/2 for example, a given increment in temperature of order e above the frozen flow value.
• Equation (26) may also be obtained from Eq. ( 6) by neglecting the reactant consumption and linearizing the Arrhenius exponent around the higher boundary temperature in the manner of Frank-Kamenetskii [17] .
• There the authors obtain the following asymptotic expression for the Ignition Damkohler number EQUATION ) where F(j8) is very accurately correlated by F(0) = 2/3 -/3 2 .
• For £ close to one, in the ignition conditions the reaction zone structure is of the same form as that of the premixed flame regime, and then the results obtained in Section 5 may also be used to determine the ignition conditions.

4. Partial burning regime

• If the activation temperature is large compared with the local temperature a very small change in this temperature below the maximum is necessary to freeze the chemical reaction.
• Only at both outer edges of the mixing layer will, again, the large reaction time be comparable with an appropriately large residence time in thick zones, where the chemical reaction will go to completion without a significant increase in temperature.
• Instead of basing this choice on the matching conditions with an asymptotic reaction zone solution for large activation energies.
• Matching to the next order, intermediate between e° and e, carried out in intermediate variables, demands the equality, at the outer edges of the reaction zone, of the slopes of the temperature distribution given by the inner and outer expansions.
• The extinction conditions cannot be determined from the analysis of this regime.

5. Premixed flame regime

• This is not the case if 3 < 1, because then the temperature reaches its maximum value within the reaction zone, and the flow outside will be chemically frozen, to all algebraic orders in T a ' 1 ; the fuel and oxidizer after crossing the reaction zone will coexist in frozen flow.
• The analysis is completely analogous, and the results may be directly written, for the case a +23 >.
• The chemical reaction will be frozen to all algebraic orders in e, because the temperature in this region will be lower than T p and the Damkohler number is not high enough to offset the effect of the Arrhenius exponent.
• Equation ( 52) is to be solved with the boundary conditions obtained from the matching conditions with the outer solutions given by Eq. ( 19) and (44).
• The partial burning analysis of the previous reaction should be used for m > 1/2. When T p becomes close to T e the concentrations of both reactants in the reaction zone are small and have relative variations of order unity; then the authors must use the analysis of the near-equilibrium, diffusion flame regime that follows.

6. Near equilibrium, diffusion flame, regime

• Finally, a diffusion controlled regime exists for which the flow is everywhere near equilibrium, so that in a first approximation the flame position and temperature distribution are determined by Eqs. (20) to (23) independently of chemical kinetics.
• This corresponds to the Burke-Schumann[l8] and classical diffusion flame analysis [19] .
• Equations ( 75) to (77) describe in the first approximation the non equilibrium effects, as dependent only on the reduced Damkohler number and y.
• The retention in Eq. ( 75) of the exponential Arrhenius factor is essential for evaluating the non equilibrium effects in the near-extinction conditions.
• There is leakage of both reactants through the flame as evidenced from the fact that (j8, -£)» and (j8i + £)_» are, functions of y and 8, different from zero.

7. Discussion and generalization

• The analysis of the previous sections covers all the regimes that can be found when analysing diffusion flames, unless they are of the unsteady or evolution type [251, or the chemical reaction is multi-step and does not admit a simplified description by a single overall irreversible reaction.
• It is interesting to observe that the parameter TJTr is the product of the nondimensional activation energy and the nondimensional heat release, or third Damkohler number, both based on the characteristic thermal energy at the temperature of the reaction zone.
• Both curves, T b (D) and T P (D), have branches that exhibit an increasing temperature with decreasing Damkohler numbers.
• The analysis is not included because this transition regime will always be unstable (as is also the case with the partial burning regime).
• Results for ignition, partial-burning and premixed-flame regimes appear, as well as a comparison of extinction Damkohler numbers with their numerical results.