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The structure of lean hydrogen-air flame balls

E. Ferna'ndez-Tarrazo, A.L. Sanchez, A. Liñan, F.A. Williams

Abstract

An analysis of the structure of flame balls encountered under microgravity conditions, which are stable due to radiant losses from H20, is carried out for -lean hydrogen-air mixtures. It is seen that, because of radiation losses, in stable flame balls the maximum flame temperature remains close to the crossover temperature, at which the rate of the branching step H + 02 -> OH + O equals that of the recombination step H + 02+M -> H02+ M. Under those conditions, all chemical intermediates have very small concentrations and follow the steady-state approximation, while the main species react according to the overall step 2H2 + O 2 -> 2H20; so that a one-step chemical-kinetic description, recently derived by asymptotic analysis for near-limit fuel-lean deflagrations, can be used with excellent accuracy to describe the whole branch of stable flame balls. Besides molecular diffusion in a binary-diffusion approximation, Soret diffusion is included, since this exerts a nonnegligible effect to extend the flammability range. When the large value of the energy of the overall reaction is taken into account, the leading-order anal-ysis in the reaction-sheet approximation is seen to determine the flame ball radius as that required for radi-ant heat losses to remove enough of the heat released by at the flame to keep the flame temperature at a value close to crossover. The results are relevant to burning velocities at lean equivalent ratios and may influence fire-safety issues associated with hydrogen utilization.

1. Introduction air. Hydrogen deflagrations persist to much lower concentrations than would be possible under The increased interest in hydrogen usage steady, planar conditions [1] because of hydrogen's focuses attention on safety concerns such as the strong propensity for diffusive-thermal instability. possibility of sustained fires developing in systems Flame balls constitute limiting configurations in having very low concentrations of hydrogen in which hydrogen , dominated in this case by its high diffusivity leading to higher flame temperatures, can occur under the leanest possible conditions [2]. Collections of flame balls can con­ tinue to burn for long periods of time in mixtures with insufficient hydrogen for ordinary flame prop­ agation [3]. For these reasons, in considering fire hazards, it is desirable to have accurate descriptions of flame-ball structures in very lean hydrogen-air gen-air combustion has been tested thoroughly [1]. mixtures. The study reported here was performed In fuel-lean hydrogen-air flame balls, the thermal in an effort to improve such descriptions and to conductivity is dominated by the values of nitrogen be better able to understand and to calculate the and oxygen, so that the approximate expression resulting flammability behavior. 5 0 7 Computational methods are sufficiently X=cp = 2:58 x 10T (r=298) : kg=(ms); (5) advanced that flame-ball structures can be suggested in [7] for methane-air flames, can be addressed numerically [4,5]. While numerical used with excellent accuracy to compute X, with approaches can include many phenomena, they the value of the specific heat at constant pressure often are not well suited to specifically identifying cp — ^2¡cPiYi evaluated with use made of NASA the most important ones. Asymptotic and analytic polynomial fits for the temperature dependence methods, on the other hand, can isolate the most of each cpi. At the same level of approximation important effects, thereby increasing understand­ [7], D¡ can be computed as the binary diffusion ing of the flame structures significantly. In addi­ coefficient of the species i into nitrogen according tion, they often can yield formulas that are readily applied to calculate quantities of interest to :7 in applications. For these reasons, in the present pD¡ — 2:58 x 10 (r=298) = ¿¡; kg=(ms) (6) work a simplified analysis is derived for better understanding of near-flammability combustion with ¿¡ = (1:11;0:30;0:83) for (O2,H2,H2O), and flame-ball burning. respectively. Thermal diffusion is negligible for all species, except for H and H2, for which the Soret factors are given by aH — —0:23 and aH2 — —0:29 [8]. A detailed description is also needed for the rate 2. Formulation of radiant energy loss per unit volume QR, which is the divergence of the radiant-energy flux vector. 2.1. Starting equations For instance, if a differential gray-gas approxima­ tion [9] is adopted, Q is a function of the mean The problem to be addressed is flame-ball R intensity of radiation / — IA=(4n) that can be ex­ combustion of lean hydrogen-air mixtures in a 4 pressed in the form QR — —aa(IA — 4(71 ), where gravity-free environment, which can be steady I A satisfies the equation and spherically symmetrical. Then the radial veloc­ ity of the fluid is zero, so that momentum conserva­ 1d r2 dI A cc (I -4oT4 tion implies that the pressure p is constant. The r2 dr a A (7) radial variation of the temperature and composi­ 3aa dr tion is obtained by integrating the conservation with boundary conditions dIA=dr — 0 at r — 0 equations for energy and chemical species and IA — 4aT^ — 0 as r —> oo. Here, a denotes the Stefan-Boltzmann constant and aa — KH2OP 1d 2 dT kr (W=WH2O)YH2O is the gray mean value of r2 dr (1) dr i the absorption coefficient corresponding to the main contributing species H O, where W — 1 d 2/d7, , 0,7, dry 2 = m¡; (2) (^¡Yi=Wi) is the mean molecular weight, com- r2 dr puted from the specific molecular weights W¡ of with boundary conditions the different chemical species i. The Plank-mean absorption coefficient K O is a function of the dr dYj H2 — == == 0 at /* == 0 (3) temperature, given for instance in [10], that in dr dr the range of temperatures of interest (270 < and T < 1300 K) can be shown to be accurately repre­ 4 sented by the expression KH20 — 5:72 x 10~ T-T x. = Y ¡ — 7,0o =0 as r —> oo; (4) (r=298) s2=kg. where the subscript oo denotes ambient quantities, The inverse of aa gives the characteristic value r is the distance to the flame-ball center, and T, p of the absorption length, which can be estimated and X are the temperature, density and thermal from the above expressions to be of the order conductivity of the gas mixture. The composition cç1 ~ 100 cm for characteristic flame-ball com­ is determined in terms of the mass fraction 7, of bustion conditions (e.g., cÇ1 — 92 cm for T — each species i, whose molecular diffusivity and en­ 1000 K, p — 1 atm, 7H2O — 0:125), a quantity that thalpy of formation are denoted by D¡ and h°, is therefore very large compared with the charac­ respectively. teristic flame-ball radius rj ~ 1 cm. According to The computation of the mass rate of production (7), in the optically thin limit of small Bouguer of species i, denoted by m¡, requires specification of numbers aarf

computing the radiant energy loss per unit rf corresponding to the stable branch away from volume; this leads to the familiar approximation the lean flammability limit. Sample profiles of

4 main species and temperature are shown in QR = 4KH2O^(^/^H2O)FH2O(T - It), (8) Fig. 3 for a flame ball with

H20 + 02 by introducing stea­ criterion (e.g., location of peak H2-consumption dy state assumptions for the intermediaries H, O, rate or maximum H-atom concentration) would OH and H02. As noted in [13], the rate (9) applies be very similar, especially for the larger values of with excellent accuracy at atmospheric and moder-

0.01 -

Fig. 1. The variation with equivalence ratio of the hydrogen-air flame-ball radius r¡ for p = 1 atm and rœ = 300 K as obtained from experiments (hollow symbols: [11]; solid symbols: [12]), from computations with 21-step chemistry (solid curve) and with one-step chemistry (dashed curve) and from the reaction-sheet analysis (dot-dahed curve). 1200

1100

1000

900 _l I I L_ _l I I L_ _l I I L_ 0.1 0.2 0.3

Fig. 2. The variation with equivalence ratio of the peak temperature rmal for the hydrogen-air flamebal l vjithp = 1 atm and Too = 300 K as obtained from computations with 21-step chemistry (solid curve) and with one-step chemistry (dashed curve) along with the results of the reaction-sheet analysis (dot-dashed curves), including the leading-order prediction f„ = T°c (thin curve) and the corrected value (thick curve).

0.5 - - 0.05 D)

0.01 0.02 r [m]

Fig. 3. Radial distributions of T/Tmíaí, YH2/YH2m, ¥02/¥02m and 7H2O as obtained from numerical integrations with 21- step chemistry for (¡> = 0.15, T&• 300 K and/; = 1 (solid curves); the dashed line shows the radial variation of the fuel mass consumption rate. ately elevated pressures, but for larger pressures are functions of order unity of the composition approaching the third explosion limit the descrip­ and temperature [13]. tion should be modified to account for H202 forma­ The (9) applies for temperatures tion from H02 and its regeneration of active larger than a crossover value Tc for which radicals. In the formulation, CM represents the effective third-body concentration, which accounts klf = cck4fCu, (10) for the non-unity third-body chaperon efficiencies whereas at temperatures T < Tc our one-step of H20 and H2 associated with reaction 4/. The approximation predicts œ — 0 due to the disap­ subscripts/and b in (9) denote forward and back­ pearance of the radicals. The value of Tc is very ward reaction-rate constants, while G, H and a dependent on the local H2 content through a, which equals one for 7Hz = 0 but decreases rapidly 7H2O = 0 as r —> oo are indicated by the dashed towards a — 1 /6 when evaluated with the small curves in Figs. 1 and 2, with the flame-ball radius nonzero values of 7H2 typically found in the reac­ in Fig. 1 defined as the location where the temper­ tion zone. Because of the relatively large activation ature reaches its crossover value. As can be seen, temperature of reaction If, r„ — 8590 K, the the resulting curves are practically indistinguish­ overall rate (9) is very sensitive to changes in tem­ able from those obtained with the 21-step mecha­ perature through the term ^/(OÍ^CM) appearing nism, indicating that the steady state is an in the cutoff factor. The associated value of the excellent approximation for all chemical inter­ dimensionless p — Taif/T + «i/ mediates. —«4/ + 1, written with account taken of the addi­ As previously mentioned, for

0.0735 there tional algebraic temperature dependences T",f exists a branch of unstable solutions, not depicted _1 and r"v of k\f and &4/CM, where ri\f — —0.7 in Figs. 1 and 2, corresponding to flame balls that and «4/ — —1.4, is found to be of order 10 at tem­ are smaller and hotter than those of the stable peratures of the order of the crossover temperature branch described herein. The flame balls along Tc. Therefore, changes in the reaction rate by a fac­ this unstable branch are sufficiently small for radi­ tor of order unity occur with small temperature ation to have a negligible effect. As the mixture increments of the order (T — Tc)/Tc ~ /T , which becomes leaner, the decreasing reaction rate are enough to accommodate the changes in burn­ requires a larger flame-ball radius to ensure an ing rate per unit surface found for increasing

, the peak temperature Tmix of from noting that dYHl + aH2YH2dT/T = T'^d the resulting solutions cannot be far from the aH crossover value (i.e., if Tmix — Tc were much larger (r 2 7H2) which allows the thermal diffusion and x molecular diffusion terms to be incorporated in a than ¡r Tc, the associated increase in chemical single Fickian-like diffusion term written as a reaction would be so large that it could not be bal­ function of the modified fuel mass fraction anced by radiation losses). Therefore, for increas­ ing equivalence ratios, the resulting flame-ball B Y={T/TOB)" *Yn2 (11) radius for the branch of stable solutions increases in such a way that radiant energy losses can bal­ and with a modified diffusivity ance the chemical heat release while keeping the a D=(T/T00)-* iDHl. (12) flame temperature at a value not far from Tc, a condition that determines at leading order the The new transport operator, which is particularly flame-ball structure, as seen below. useful for the analysis of convection-free steady configurations, can be used in (2) to write 1 d /p£>o2j2d702\ 1 d /p£>H2o;,2dyH2o\ 1 1 3. The thin reaction-layer description r àr V W0l dr ) r dr KlW^o dr ) _ 1 d f pD 2dY\ For large nondimensional activation r ~7dr \2WH2 ~dr) j3> 1, the reaction occurs, as shown in Fig. 3, = w (13) in a thin layer, where the fuel mass fraction and the temperature variations from the crossover for the stoichiometry of the global reaction value are of order 7H2/7H2OO ~ (T - Tc)/Tc ~ 1 2H2 + 02 -> 2H20, whereas the energy equation /T > 3.1. Relationships between the chemical species 5 where q = -h°Hi0/WH2 = 1.21 x 10 kj/kg is the For simplicity, we shall use the assumption, amount of heat released per unit mass of fuel con­ not strictly necessary, of equal temperature depen­ sumed. Results obtained with this one-step dence of the diffusivities D, D0l and Z>H2O- Then, description by integrating Eqs. (13) and (14) with the first two equations in (13) can easily be inte­ boundary conditions dl' ¡dr — d7,/dr = 0 at grated, with the boundary conditions (3) and (4), r = 0 and T - rœ = 7 - 7H, = YQl - 70, = to yield the two simple relations WHlo D0l temperature dependence of the transport coeffi­ YH2O — 2 W0l DHlo cients are v = 0.85 and y — 0.99 as dictated by Eqs. (5), (6) and (12), with cp assumed in Eq. (5) WHl0 D y H2oo - Y) (15) to be only a function of the temperature with a WHl DHl0 0AS weak dependence cp ex T , as corresponds giving the mass fractions of water vapor and oxy­ approximately to air. Here, gen in terms of the modified fuel mass fraction Y. 2 2Wn2r f6] In this approximation, the effect of thermal diffu­ a> and sion is incorporated by using in evaluating Eq. a (pD^Yn, (15) an averaged increased diffusivity D/DHl — I V IYH2J : -" 1.154, yielding (20) e =Tool-oo/{pD) a YH2pL=Wm_D_ = V are the dimensionless reaction rate and its corre­ YH^ WH, DH,O ' sponding heat release. In the dimensionless for­ and mulation, the original unknowns Tmxx and r/ have been replaced with the ambient-to-fiame Yo2r = \WQl D YHlco temperature ratio Y0lco 2 WHl D0l Y0lco "co ^ co / * m (21) = 1-0/0.234, (17) and with the value of the ratio of the characteristic for the water vapor and oxygen mass fractions in values of the local volumetric heat losses by radi­ the reaction zone, obtained from (15) with Y = 0. ation and conduction in the hot flame region In view of this last equation, it is clear that the lean regime of flame balls is restricted to configu­ 8 = 4KH20^(^air/^H2o)FH2oXa*'/A., (22) rations with 0, a result of the differential diffusion first the reaction-zone values KH2O, and Ar evaluated pointed out by Joulin [15]. Note that this limiting at T — T , and with the constant molecular equivalence ratio, corresponding to the effective mix weight of air W.¿r replacing W, a reasonably accu­ stoichiometric conditions for which both reactants rate approximation for the lean conditions ex­ are simultaneously depleted in the reaction sheet, plored herein. Eqs. (18) and (19) are to be is clearly affected by thermal diffusion, which, integrated with boundary conditions #'(0)=y according to the approximation D/Du2 — 1.154 (0) = 0 and 0(oo) - 0*, = y(oo) -1=0. The employed in evaluating the water vapor and oxy­ additional conditions that the peak value of 0 be gen at the flame, increases the transport rate of 0 = 1 and that 0 = (rc/roo)0oo at r\ = 1 are hydrogen by approximately 15%, thereby also needed to determine the unknowns 0 and e. influencing the corresponding limiting value at œ the lean fiammability limit. For larger equivalence ratios

0.234, oxygen becomes the deficient 3.3. Solution with the reaction-sheet approximation reactant at the reaction layer. Then, because of For large values of the nondimensional activa­ the lower molecular diffusivity of 02, the flame- ball temperature drops rapidly from the value tion energy /?, the reaction-rate terms in (18) and achieved when the much more diffusive species (19) can be replaced by dirac-delta functions H is the limiting reactant, as shown in Fig. 2. located at r\ — 1 that cause the gradients of tem­ 2 perature and reactant to have a discontinuity at the flame according to 3.2. The dimensionless formulation 0f- - cf+ = ey+. (23) For given ambient conditions, the computa­ The inner-side gradient y1, can be neglected tion of Tmix and rf requires integration of the last equation in (13) together with the energy equation above since y remains constant for r\ < 1. In fact, (14) with the boundary conditions given above. it takes a value y — 0, because the temperature The problem can be formulated in dimensionless gradient on the inner side of the flame sheet form by scaling the variables according to 0'(1) — 9'f_ is not large enough to freeze the reac­ 6 = T/T , y = Y/Y and r¡ = r/r to give tion, so that for 0 < r\ < 1 the mixture is in near max Hlco f equilibrium with y — 0. Integration of Eq. (18) for 0 < r\ < 1 with Q — 0 and y — 0, and with (18) r\L 0 boundary conditions 0'(O) — 0 and 0(1) — 1 gives the temperature distribution inside the flame ball, ¿(0yy)' = fi, (19) including the value of &._. Similarly, integration of Eqs. (18) and (19) with Q = 0 for r\ > 1 with with the prime / denoting differentiation with re­ boundary conditions 0(1) — 1 — y(l) —0 and spect to the coordinate r\. The exponents for the 0(oo) — 0œ — y(oo) — 1=0 determines the distri- butions of 9{r¡) and y{r¡) in the outer frozen tions of water vapor and oxygen given in Eqs. region, including the gradients of temperature (16) and (17). In the integration, the temperature and fuel mass fraction outside the flame must be computed in terms of Y as dictated by 9'{\) — 0f+ andy(l) ~y'f+, whose variation with s as well as that of 91, are given in Fig. 4 for two l-0 = Qy-fff_(t,-l), (24) representative values of 9^. obtained by integrating twice the chemistry-free For p ;» 1, in the leading-order approximation equation obtained by adding (18) to the product corresponding to the limit p —> oo, the reaction of (19) times Q. The solution is greatly simplified layer appears as infinitesimally thin, and the peak when the effect of heat losses is neglected by dis­ temperature and the crossover temperature take carding the last term in (24), an approximation the same leading-order value Tc — Tmix — T°c, that is justified by the relatively small values of 1 where T°c is the crossover temperature obtained 9 , seen in Fig. 4. Under those conditions, inte­ for a given equivalence ratio

in computing the gradients 9' _, 9l , and yl as f f+ f+ where >¿ , r/, and 9^ can be evaluated in terms of functions of e, to be substituted into (23) to provide + the leading-order results, and the integral on the an implicit relation that can be solved for e. These right-hand side can be performed with use made leading-order results can be used in (22) to deter­ of (24) expressed in the simplified form mine the corresponding flame-ball radius r/, which 1 — T/T — QY/Yu - Solving simultaneously is plotted in Fig. 1. mxx 2co 1 the three Eqs. (10),°° (25) and 1 - Tc/Tmax = Relative corrections to rmax, of order /T , can QYc/Yu2co determines the corrected values of be obtained by analyzing the quasi-planar reac­ Tmix and Tc as well as the fuel mass fraction at tion layer; the analysis will also provide the value _1 the crossover point Yc. The resulting value of of the fuel mass fraction Yc ~J5 FH2OO at the Trnax is shown in Fig. 2 as a thick dot-dashed curve. crossover point where reaction (9) freezes, to be As can be seen in Figs. 1 and 2, the analytical employed in evaluating the function a < 1 in solution describes with remarkable accuracy the (10), which in turn determines the corrected cross­ whole branch of stable solutions. Errors in the over temperature Tc. The amount of fuel burnt flame-ball radii are typically smaller than 10%. per unit time can be determined by integrating On the other hand, the leading-order prediction (19) across the reaction layer with the reaction for the peak temperature Tmax — T°c contains rate Q evaluated with the equilibrium mass frac- errors on the order of 100 K, but these errors reduce considerably when the first-order correction is introduced by accounting for the solution within the reaction layer, yielding overpredictions of peak temperatures that are always below 30 K. Note that the approximations used for the analytical results necessarily limit their accuracy, both for small flame balls approaching extinction, where the reaction region can be anticipated to extend all the way to the center of the flame ball, thereby invalidating the flame-sheet approximation, and also for very large flame balls with significant radi­ ation; these are such that the temperature gradient appearing on the inner side of the flame sheet may quench the chemical reaction there, producing sig­ nificant fuel leakage, a higher-order effect not accounted for in the present analysis, which neglects the last term in Eq. (24). The associated departures are however small, so that the reac­ tion-sheet description is satisfactory, as demon­ strated by the plots in Figs. 1 and 2.

4. Conclusions Fig. 4. The variation with E of y',+, SL_, and SL+ for floo = 0.25 (solid curves) and 6^ = 0.30 (dashed curves) as obtained from integrations of Eqs. (18) and (19) as While steady, freely propagating planar flames described in the text. in hydrogen-air mixtures at normal atmospheric conditions exist only for equivalence ratios above a [2] F.A. Williams, J.F. Grcar, Proc. Combust. Inst. 32 lean flammability limit . increases in such a way that radiant heat losses keep [7] M.D. Smooke, V. Giovangigli, in: M.D. Smooke the peak temperature at a value that barely exceeds (Ed.), Reduced Kinetic Mechanisms and Asymptotic Tc. Under those conditions, the one-step chemistry Approximations for Methane-Air Flames, Lecture recently developed for near-limit hydrogen-air Notes in Physics, vol. 384, Springer-Verlag, Berlin, deflagrations [13,14] can be used to describe with 1991, pp. 1-28. excellent accuracy the whole branch of stable flame [8] R.M. Fristrom, L. Monchick, Combust. Flame 71 (1988) 89-99. balls. As may be seen in Fig. 1, while different the­ [9] W.G. Vincenti, C.H. Kruger, Introduction to Phys­ oretical results agree well with each other, experi­ ical Gas Dynamics, Wiley, New York, 1967, pp. mental results for flame-ball radii exhibit a 491^195. considerable amount of scatter, possibly due to [10] R.S. Barlow, A.N. Karpetis, J.H. Frank, J.-Y. unsteady effects in the flame-initiation stage. Chen, Combust. Flame 127 (2001) 2102-2118. Further well-controlled microgravity flame-ball [11] P.D. Ronney, K.N. Whaling, A. Abbud-Madrid, J.L. experiments therefore would appear to be Gatto, V.L. Pisowicz, AIAA J. 32 (1994) 569-577. worthwhile. [12] M-S Wu, J-B. Liu, P.D. Ronney, Proc. Combust. Inst. 27 (1998) 2543-2550. [13] D. Fernández-Galisteo, A.L. Sánchez, A. Liñán, Acknowledgements F.A. Williams, Combust. Flame 156 (2009) 985- 996. [14] D. Fernández-Galisteo, A.L. Sánchez, A. Liñán, This work was supported by the Spanish F.A. Williams, Combust. Theoret. Model. 13 (4) MCINN through Project # ENE2008-06515 and (2009) 741-761. by the Comunidad de Madrid through Project # [15] G. Joulin, SIAM J. Appl. Math. 47 (1987) 998- S2009/ENE-1597. 1016.

[1] P. Saxena, F.A. Williams, Combust. Flame 145 (2006) 316-323.