The upward propagation of a lean methane-air ﬂame front in a v ertical tube is investigated. The velocity of the ﬂame front has been measured with an ar- ray of photodiodes set along the tube wall and, independentl y, from photographic records. A PIV system triggered by a photodiode signal has bee n used to measure the velocity of the ﬂow induced by the ﬂame front in a vertical plane through the axis of the tube. The contour of the luminous region of the ﬂ ame front, as- sumed cylindrically symmetric, has been extracted from the rec orded images. As expected, the shape and velocity of a very lean ﬂame front, and the velocity of the fresh gas relative to the front, are similar to those of a bubb le rising in the tube. The ﬂow of the burnt gas features a region of low velocity (rel ative to the ﬂame front) which enhances radiation losses and seems to play an i mportant role in the extinction of the ﬂame at the ﬂammability limit. This limit is found to depend very sensitively on the temperature of the tube wall. A simple mod el is proposed of the ﬂow around the axis of the tube and the combustion around the ti p of the ﬂame front. This model uses the measured gas velocity at the axis of the tube together with simpliﬁed conservation equations to compute the temper ature and species concentrations along the axis for a given kinetic scheme and ra diation law. The results for a single overall Arrhenius reaction and for a fou r-step reduced scheme, both in an optically thin gas, are in reasonable agreement wi th our experimental data and shed some light on the roles played by radiation loss es and kinetic eﬀects on the ﬂammability limit measured in the standard ﬂammabili ty tube. The model can also be used to test other kinetic schemes and radiation la ws. Keywords Lean premixed ﬂames ·Upward propagation ·Flammability limit. 1 Introduction The upward propagation of a premixed ﬂame front in a long verti cal tube which is open at the bottom, closed at the top, and ﬁlled with a very lea n reactive mixture E. T. S. Ingenieros Aeron´ auticos, UPM, Plaza Cardenal Cisneros 3, 280 40 Madrid, Spain E-mail: fhiguera@aero.upm.es 2 V. Muntean, F. J. Higuera is a much studied problem. It arises in connection with the st andard ﬂammabil- ity tube ﬁrst proposed by Coward and Jones [1] as a means to cha racterize the ﬂammability of gas mixtures. Buoyancy plays a dominant role in the motion of these ﬂame fronts. Levy [2] noted that for most fuels, hydrogen excepted, the shape and velocity of near-limit ﬂame fronts are similar to those o f a bubble rising in a tube ﬁlled with a liquid, as determined by Davies and Taylor [3]. Extinction at the ﬂammability limit begins at the tip of the ﬂame front, and the ﬂammability limit depends on the radius of the tube. Thus, in their experiments w ith ammonia-air and propane-air mixtures, Babkin et al. [4] found that the mini mum possible con- centration of the reactant that is depleted by the ﬂame incre ases with the radius of the tube for lean ammonia-air mixtures and rich propane-ai r mixtures, while it decreases with the radius of the tube for rich ammonia-air mi xtures and is nearly independent of the radius for lean propane-air mixtures. Att empts at explaining extinction in terms of ﬂame stretch date back to Lewis and von E lbe [5], while von Lavante and Strehlow [6] numerically found that the stre tch is dominated by the strain rate of the ﬂow and is of the right order to aﬀect the inner structure of the ﬂame (see Ref. [7] for a more recent computation). Howeve r, the strain rate of the buoyancy-dominated ﬂow scales with the inverse of the s quare root of the tube radius (see, e.g. Ref. [8]), which suggests that, in som e cases, when the Lewis number of the reactant that is depleted by the ﬂame is above un ity, the concen- tration of this reactant at the ﬂammability limit should inc rease when the radius of the tube decreases (see, e.g. Buckmaster and Mikolaitis [8 ] and Williams [9]). This is contrary to some of the results of Refs. [1] and [4], am ong others. The issue was taken up by Shoshin et al. [10], who explained observed tre nds by the fact that the coupled eﬀect of positive ﬂame stretch and preferen tial diﬀusion locally strengthens the tip of the ﬂame in mixtures with Lewis number ( of the deﬁcient reactant) below unity, and weakens it in mixtures with Lewis n umber above unity. These authors concluded that ﬂame stretch and preferential diﬀusion alone cannot always account for the observed fact that extinction of a limit ﬂame begins at its tip. Later, Shoshin and Jarosinski [11] and Shoshin et al. [12 ] proposed that heat losses from the ﬂame due to the eﬀect of radiation in a low velo city region that appears below very lean ﬂames may explain the observed extinct ion. In this paper, we report an experimental investigation of ver y lean methane-air ﬂame fronts propagating upward in a tube. The velocity of the ﬂame front and the velocity ﬁeld of the ﬂow it induces in the tube are measured us ing photographic and PIV techniques. A simple model of the ﬂow around the tip of t he ﬂame front is proposed that uses the measured velocity of the gas together with approximate en- ergy and species conservation equations to compute the tempe rature and species concentration proﬁles along the axis of the tube. Assuming a f our-step reduced scheme, an optically thin gas, and transparent tube walls, t his model gives predic- tions for the ﬂammability limit that are in good agreement wi th our experiments, and partially supports the view that radiation losses are res ponsible for extinction in the standard ﬂammability tube. The model can also be used a s a workbench to test more realistic chemical kinetics and heat losses. Upward propagation of very lean methane-air ﬂames in vertical tubes 3 Seeding system PIV camera FlameLaser sheet CH4 CO2N2O2Laser sync Flame's tip Δt Fig. 1Sketch of the experiment 2 Methods 2.1 Experimental setup and techniques Figure 1 is a sketch of our experimental setup. The central ele ment is a vertical quartz tube of 54 mm internal diameter, 3 mm wall thickness, and 1.9 m long, whose lower end is open to the atmosphere. In each run, the tub e is ﬁlled through a valve at its top with a mixture of methane and air of chosen equi valence ratio, which is prepared from 99.5-pure methane, oxygen and nitrogen , whose ﬂow rates are measured separately using EL-Flow Bronkhorst mass ﬂow co ntrollers. The accuracy of these controllers is 0.5% of the reading plus 0.1 % of the apparatus’ span. However, the ﬂow rates of oxygen and nitrogen are set to th e constant values 2.52 l n/min and 9.48 l n/min, respectively, the same in all the experimental runs, which are independently measured with an accuracy of 0 .5% by means of Ritter TG10 Drum-type gas meters used to calibrate the oxygen an d nitrogen Bronkhorst controllers at ﬂow rates near those used in our expe riments. (Here lndenotes liters referenced to normal conditions; a temperat ure of 0◦C and a pressure of 101325 Pa). The ﬂow rate of methane changes from r un to run, as this is the means used to change the equivalence ratio of the mixture . However, these changes are small, the methane ﬂow rate being in the range fro m 0.7 l n/min to 0.8 ln/min, which is about a 2% of the capacity of the methane contro ller. The mid value of this range is independently measured with an accura cy of 0.2% by means of the Ritter TG05 gas meter used to calibrate the methane con troller. Thus it is expected that the combination of calibration and a reduced ap paratus span will decrease the error of the methane ﬂow rate measurements to ab out 2%×0.6% = 0.012% of the full span. The typical error of the equivalence r atio is then 1%. Error bars of this variable are included in Fig. 2 below. 4 V. Muntean, F. J. Higuera The fresh mixture ﬂows through the top of the tube with a veloci ty of about 0.1 m/s for at least three minutes, to evacuate the burnt gas f rom previous runs. The volume of mixture used amounts to 10 times the volume of the tube, and no diﬀerence was observed if the fresh gas is kept ﬂowing longer or if this gas is left in the tube for some time after closing the valve. A CO 2probe shows that the concentration of this species left in the tube from previous r uns rapidly decreases when the fresh mixture begins to ﬂow. After the valve at the top is closed, the lower 15 cm of the tube are ﬁlled with an enriched mixture through a second valve that opens at this h eight. The purpose of the enriched mixture is to facilitate ignition and to smoot h the transition of the ﬂame to the region of the tube ﬁlled with the mixture of interes t. The device has been used by others before (e.g., Levy [2], Shoshin et al. [ 10], Shoshin and Goey [13]). When the ﬁlling process is complete, the mixture i n the tube is allowed to settle for about two minutes and is ignited with an electri cal resistance near the lower end of the tube. Transition from the enriched mixture to the test mixture of interest occurs in a region that may extend somewhat above the lower valve, due to the turbulence and diﬀusion that act following the ﬁlling of the lower part of the tube. As in Ref. [13], where the use of an enriched mixture t o launch the ﬂame is discussed in some detail, the length of this transition re gion is judged by the distance between the lower valve and the location at which th e ﬂame approaches quasi-steady shape and speed. As in Ref. [2], the ﬂame was obse rved to fail at most a few tube diameters above the lower valve when the upper part of the tube is ﬁlled with a non-ﬂammable mixture. These tests suggest tha t inhomogeneity of the gas mixture due to the enriched mixture present at the lower part of the tube does not extend to the test region where the measurements desc ribed below are made. The toroidal eddy found in Ref. [10] for a 24 mm tube, whi ch rises with the ﬂame and carries gas from the lower to the upper part of the tube, was not observed in our experiments, but see further comments on this point in section 3. The upward velocity of the ﬂame front (but not its shape) was m easured by two means. On the one hand, an array of photodiodes was set alo ng the tube which are sequentially excited by the passage of the ﬂame, despite it s weak luminosity. On the other hand, the motion of the ﬂame front was recorded by a video camera that images a region of the tube about 40 cm long. The ﬂame velo cities measured by the two methods are in good agreement and show that the ﬂame rises steadily after an initial transient. Theshapeoftheﬂamefront(assumedcylindricallysymmetric) wasdetermined from the images of a second camera ﬁtted with a large aperture objective (Tamron, 60 mm) that images a shorter region of the tube of about 68 mm wi th a resolution of 40–70 pixel/mm and is triggered once per run by the signal of a photodiode that detects the passage of the ﬂame at a lower section of the tube. Tests have shown that an exposure time of 1.5 ms with a f-number of 2.0 allows to g ather enough light from the faint ﬂame to clearly detect its contour witho ut causing blur. The upward velocity of the ﬂame front is about 23 cm/s (see Fig. 2 b elow) and the thickness of the unstretched ﬂame is about 1.5 mm. The Canny al gorithm [14] with a Laplacian of Gaussian ﬁlter was used to extract the cont our of the ﬂame front from the recorded images. A PIV system was used to measure the velocity of the gas around t he ﬂame front in a vertical plane through the axis of the tube. The PIV c amera (nac Mem- recam HX-3) images the same region of the tube as the camera us ed to determine Upward propagation of very lean methane-air ﬂames in vertical tubes 5 the contour of the ﬂame with a resolution of 32 pixel/mm. The tw o cameras and the pulsed Nd:YAG laser of the PIV system were triggered by the same photodiode signal. The gas mixture was seeded with alumina particles, wh ich were produced in a ﬂuidized bed inserted in the gas line some distance upstr eam of the valve at the top of the tube. The density of seeding was tuned to optimi ze the quality of the PIV correlations by changing the fraction of gas that bypa sses the ﬂuidized bed. This optimization was not always possible because the th ermal expansion of the gas across the ﬂame drastically changes the seeding dens ity, and sometimes makes it necessary to use diﬀerent seedings to determine the velocities upstream and downstream of the ﬂame in diﬀerent runs. Seeding the gas with solid particles may have an inﬂuence on r adiation losses, whichcouldsigniﬁcantlyaﬀecttheexperimentasradiationl ossesfromthecombus- tion products already seem to play an important role in the ext inction of the ﬂame at the ﬂammability limit. The properties, conditions and nu mber density of the seeding particles arenotaccurately known,which makesitdi ﬃcult todirectlyeval- uate radiation losses from these particles. However, ﬂame f ront speeds measured for nominally identical cases with and without particles, a nd the ﬂammability limit determined with and without particles, coincide within the experimental uncer- tainty, which suggests that radiation from the seeding part icles has no signiﬁcant eﬀect on the ﬂame at the density of seeding used. Thermophoreticforcesontheseedingparticlesareacauseo fsystematicerrorin LDV and PIV measurements in regions with strong temperature gradients (Talbot et al. [15], Beresnev and Chernyak [16]). Pending a comprehe nsive assessment of the eﬀect of thermophoresis in our experiment, preliminary t ests with various particle sizes suggest that thermophoresis does not drasti cally aﬀect the measured gas velocity within the ﬂame when the seeding particles are l arger that a few microns. Thetemperatures ofthe tube and thegas weremeasuredat diﬀe rent sections of the tube with type-T (Cu–CuNi) thermocouples. These measure ments have been used to study the nonuniform cooling of the tube following th e passage of a ﬂame, and to determine the minimum time required between runs. It wa s found that temperature variations of a few degrees may have an eﬀect on t he propagation of near limit ﬂames. 2.2 Approximate model A simple model is set up to approximately describe the ﬂow and t he near limit ﬂame in a region around the axis of the tube, where extinction ﬁr st occurs at the ﬂammability limit. The model relies on a combination of e xperimental results and simpliﬁed conservation equations. It gives reasonable r esults and serves as a workbench where diﬀerent kinetic and heat losses models can be compared and tested. Previous work (Peters and Rogg [17]) has shown that a four step reduced chemical kinetics mechanism with three intermediate specie s out of steady state suﬃces to describe most aspects of lean premixed ﬂames. Here, following S´ anchez et al. [18], we choose OH as the radical representative of the H2–O2radicals pool and take CO, H 2and OH as the three intermediate species out of steady state. 6 V. Muntean, F. J. Higuera This yields the four-step reduced scheme CH4+2OH→CO+H 2O+2H 2 (1) H2+O2⇌2OH (2) 2OH+H 2⇌2H2O (3) CO+H 2O⇌CO2+H2 (4) with rates (moles per unit volume per unit time) ω1= (k11[H]+k12[OH])[CH 4] (5) ω2=k2f[O2][H]−k2b[O][OH] (6) ω3= (k3f[O2][H]−k3b[HO2])[M] (7) ω4=k4f[CO][OH] −k4b[CO2][H] (8) where [i] denotes the molar concentration of species i, [M] = 0 .6[H2] +0.4[N2] + 0.44[H2O]+0.4[O2]+1.5[CO2]+3[CH 4]+[OTHERS] is the eﬀective third body concentration, and the molar concentrations of H, O and HO 2that appear in (5)–(8) are given by [H] =K1[H2][OH] [H2O],[O] =K2[OH]2 [H2O], [HO2] =k3f[M][H]+ k5b[H2O] k3b[M]+k5f[OH][O2],  (9) which are simpliﬁed forms of more complex relations derived from the conditions of partial equilibrium and steady state of some reactions and sp ecies (see Ref. [18] for details). The reaction constants coincide, with minor chan ges, with those used in Ref. [18] to compute the structure and speed of planar ﬂames a t high temperature and pressure. Consider the stationary ﬂow of the gas at the axis of the tube in a reference frame moving with the ﬂame front. Approximate energy and speci es conservation equations along the axis read ρvcpdT dx=d dx/parenleftbigg λdT dx/parenrightbigg +αλTb−T R2+WT−L, ρvdYi dx=d dx/parenleftbiggλ/cp LeidYi dx/parenrightbigg +Wi,(10) wherexis downward vertical distance; v(x) is the downward velocity of the gas relative to the ﬂame; ρandTare the gas density and temperature; Yi= Mi[i]/ρand Le iare the mass fractions and Lewis numbers of the seven species i= O2,CH4,CO2,H2O,CO,H2and OH, with Midenoting the molecular masses of these species; cpandλare the speciﬁc heat and thermal conductivity of the gas; and the reaction and radiation terms WT,LandWion the right-hand sides of the equations will be discussed below. The following approximati ons have been made to write these equations: ( i) Radial diﬀusion of species and heat are neglected in the transport region of the ﬂame, which seems reasonable whe n the thickness of this region is small compared to the curvature radius of the ﬂ ame front; ( ii) radial diﬀusion of species is neglected downstream of the ﬂame, whe re the gradients of Upward propagation of very lean methane-air ﬂames in vertical tubes 7 the species concentrations are small; ( iii) radial heat conduction is approximated byαλ(Tb−T)/R2downstream of the ﬂame, where Tbis the ﬁnal combustion temperature, Ris the radius of the tube, and α= 15 (α= 0 upstream of the ﬂame). This αis a model parameter, and its value is chosen for the results t o ﬁt a limited number of axisymmetric simulations carried out wi th the full mass, momentum and energy conservation equations. Equations (10) a re to be solved together with the equation of state /summationdisplay iYi MiρT=/summationdisplay iYiu MiρuTu (11) and the boundary conditions (in terms of the equivalence rati oφ) T−Tu=YCO2=YH2O=YCO=YH2=YOH= 0 YCH4=YCH4u=φ 17.167+φ YO2=YO2u=4 17.167+φ(12) forx→ −∞, with the subscript udenoting conditions of the fresh gas far above the ﬂame front, and dT dx=dYi dx= 0. (13) forx→ ∞. The production rates on the right-hand sides of (10) are WT=q1ω1+q2ω2+ q3ω3+q4ω4,withq1= 356.082kJ/mole, q2=−78.852kJ/mole, q3= 562.043kJ/mole, andq4= 41.114 kJ/mole, WO2=−MO2ω2,WCH4=−MCH4ω1,WCO2= MCO2ω4,WH2O=MH2O(ω1−ω4),WCO=MCO(ω1−ω4),WH2=MH2(2ω1− ω2−ω3+ω4), andWOH=MOH(−ω1+ω2−ω3). Since nitrogen is by far the most abundant species, only binary diﬀusion of each species in nitrogen is con- sidered, with constant Lewis numbers LeO2= 1.05, LeCH4= 1, LeCO2= 1.39, LeH2O= 0.83, LeCO= 1.07, LeH2= 0.3, LeOH= 0.74, and a thermal conductivity λ=λu(T/Tu)κwithκ= 0.75 andλu= 2.53×10−2N s−1K−1. The last term on the right-hand side of the energy equation in ( 10) represents the radiation losses, L= 4σK/parenleftBig T4−T4 u/parenrightBig with K=ρR0T/parenleftbigg KCO2YCO2 MCO2+KH2OYH2O MH2O/parenrightbigg ,  (14) where an optically thin gas and transparent tube wall are ass umed. Here σand R0are the Stefan-Boltzmann constant and the universal gas con stant, and the Planck mean absorption coeﬃcient Kis taken from Ju et al. [19], who evaluated it with the statistical narrow band model (Fiveland [20], So uﬁani and Taine [21]). We assume, in addition, that radiation is due only to the reac tion products CO 2 and H 2O. The factors TKCO2andTKH2Odepend only weakly on temperature. Computations carried out with the four step reduced kinetics for planar ﬂames without radiation losses (1 /R=L= 0,ρvuniform) show that the solution of (1)–(13) almost exactly reproduces the ﬂame speed of the GRI- Mech 3.0 scheme 8 V. Muntean, F. J. Higuera implementedintheCanterasoftware(seewww.cantera.org) forvaluesoftheequiv- alence ratio below 0.7, and gives proﬁles of temperature and the seven species whose evolution is computed that are in good agreement with t hose derived from the complete kinetic scheme. A simpler kinetic scheme has been also used consisting of a sin gle overall Ar- rhenius reaction that gives WT/q=WCO2/(11/4) =WH2O/(9/4) =−WCH4= ρYCH4Bexp(−Ta/T), with activation temperature Ta= 18750 K (in line with Ref.[22]),frequencyfactor Bchosentohaveaplanarﬂamevelocity UL= 4.63cm/s forTu= 300 K and YCH4u= 0.03 (see, e.g. Ref. [23]), and q= 5.01×104kJ/kg. A momentum equation would be needed to close the problem and co mpute the velocity v(x), which in turn would require information on the pressure at t he axis of the tube. Here this equation is replaced by the experimen tally measured velocity proﬁle. For the numerical treatment, Eqs. (10) are discretized using ﬁnite diﬀerences on a uniform grid of 1000 points in the axial range of Fig. 3(b) b elow, which gives a grid spacing of 0.05 mm. The position of the ﬂame relat ive to the veloc- ity proﬁle is given by choosing the point, say x0, where the scaled temperature θ= (T−Tu)/∆T, withcp∆T= (q1+2q2+q3+q4)YCH4u/MCH4, takes a cer- tain value, typically 0.8, and the equivalence ratio in (12) is iteratively computed to satisfy this condition. The point x0is then changed and the computation is repeated until the mass ﬂux ρvvaries smoothly across the ﬂame. The value of φ thus determined is compared to the known equivalence ratio for the experimental run from which the velocity proﬁle was extracted. The diﬀeren ce is a measure of the accuracy of the method and the simpliﬁed kinetic and heat l osses in (1)–(8) and (14). 3 Results and discussion The recorded shape of the ﬂame front consists of a rounded cap similar to a bubble rising in the tube followed by a long skirt; a shape ofte n reported in the literature. Inspection ofa number of ﬂames shows that, in mo st cases when ignition isproducedsymmetricallyandcareistakentominimizeresid ualmotionsofthegas in the tube and thermal perturbations through the tube wall, the ﬂame front tends to rise steadily and to take a nearly axisymmetric form after an initial transient. Asymmetries in the ﬂame skirt near the wall could not be avoided . The velocity of the ﬂame front, U0, is shown in Fig. 2 as a function of the equivalence ratio, φ. Values of the velocity obtained from the photographic reco rds and from the array of photodiodes coincide within the experim ental error. The data in Fig. 2 are averages over at least ﬁve runs for each valu e of the equivalence ratio. Also shown in this ﬁgure is the ratio of the average len gth of the skirt, H, measured on the recorded images, to the radius of the tube, R. For the lean mixtures considered here, the velocity U0is nearly independent of the equivalence ratio and close to the velocity of a bubble r ising steadily in the tube. For a tube of 54 mm and a heavy-to-light ﬂuid density ratio of 5, which is about the burnt-to-fresh gas temperature ratio for φin the range of Fig. 2, this velocity is UDT= 22 cm/s, from Davies and Taylor formula [3]. The velocity of th e ﬂame relative to the fresh gas, ufsay, should be of the order of the unstretched ﬂame velocity in the fresh mixture. This is small compared to U0, of the order of a Upward propagation of very lean methane-air ﬂames in vertical tubes 9 19 20 2 21 22 3 23 24 4 25 0.52 0.54 0.56 0.58 0.60 0.62 H/RU0(cm/s) φ Fig. 2Upward velocity of the ﬂame front, U0(red), and ratio of the length of the ﬂame to the radius of the tube, H/R(green, right-hand side scale), as functions of the equivalence ratio of the mixture, φ. The values shown are averages over several ﬂames for each equivalence ratio. The error bars show the dispersion of the results, which increases w henφdecreases. The minimum φin the ﬁgure is slightly above the ﬂammability limit that we ﬁn d when the tube is allowed to cool down between runs. (b)(a) Fig. 3 (a) Velocity of the gas relative to the ﬂame (arrows), contours of const ant vorticity (dashed), and contour of the luminous region of the ﬂame (solid), f orφ= 0.53. (b) Velocity relative to the ﬂame front along the axis of the tube for φ= 0.55 (solid), 0.54 (dashed), and 0.53 (dotted). Diﬀerent proﬁles, corresponding to diﬀerent runs, are sho wn for each value of φ to give an idea of the error of the measurements. The origin of vertical d istance is set at the local velocity maximum. 10 V. Muntean, F. J. Higuera few centimeters per second, though it increases rapidly wit hφ. As a consequence, the tip of the ﬂame front is conﬁned to a neighborhood of the st agnation point of the equivalent bubble ﬂow, and the rest of the ﬂame front lies n ear the streamline acrossthispoint.Thedecreaseof H/Rwithincreasing φisinqualitativeagreement with the estimate H/R∼U0/ufderived from mass conservation. The small increase of U0for the two leftmost points of Fig. 2 may be an eﬀect of the ignition conditions which is brought to bear by the small velocity of the burnt gas relative to the near limit ﬂame (see below). Because of th is small velocity, some of the gas from the bottom of the tube may accompany the ﬂa me during a signiﬁcant part of its evolution and, since this gas carrie s part of the excess of energy used for ignition, it mildly aids ﬂame propagation. T he eﬀect is inherent to the structure of the ﬂow behind a near limit ﬂame. It may par tially account for the sensitivity of the ﬂammability limit to experimental cond itions [1]. Figure 3(a) shows a sample velocity ﬁeld of the gas relative t o the ﬂame front. This velocity is obtained subtracting the velocity of the fr ontU0from the velocity of the gas relative to the tube, measured with PIV. The dashed curves in Fig. 3(a) are vorticity contours. Also included in this ﬁgure is the co ntour of the luminous region of the ﬂame (solid curve), which has been extracted fro m the images of the ﬂame and ﬁtted using least squares splines. The downward velocity of the gas relative to the ﬂame at the axi s of the tube is shown in Fig. 3(b) for three values of the equivalence ratio . Diﬀerent proﬁles for the same value of the equivalence ratio correspond to diﬀe rent experimental runs and give an idea of the magnitude of the experimental erro r. As can be seen, the velocity of the gas ﬁrst decreases on approaching the ﬂam e front from above, as it would do for a rising bubble; then increases on crossing the ﬂame front, due to the gas thermal expansion; decreases again downstream of t he ﬂame, due to the buoyancy force acting on the burnt gas; and ﬁnally increa ses at some distance downstreamoftheﬂamefront,duetotheconvergenceofthebu rntgasthatcrossed the ﬂame at diﬀerent points of the front (see velocity arrows in Fig. 3(a)). The velocity of the gas ahead of the ﬂame front changes little when the concen- tration of methane is changed. However, the velocity of the b urnt gas undergoes more important changes. A region of low velocity is apparent in Fig. 3(a) down- stream of the ﬂame, and the size of this region increases when the methane con- centration decreases. Recirculation in the reference fram e tied to the ﬂame front was seldom observed in these experiments, though it has been o ften observed in experiments with narrower tubes [10]. In any case, the existen ce of a low velocity region reﬂects a competition between buoyancy, which pushe s the hot gas upward around the axis of the tube, and the downward ﬂow that emerges f rom the ﬂame front with a velocity of order ub=ρuuf/ρb(from mass conservation across the ﬂame, with ρuandρbdenoting the densities of the fresh and burnt gases). The latter ﬂow has no analogue for a rising bubble. The pressure v ariations arising in this ﬂow are of order ∆pb=ρbu2 b, while the pressure variations in the fresh gas above the ﬂame front are of order ∆pu=ρuU2 0. Bothuf/U0and the ratio ∆pb/∆pu=O/bracketleftbig (ρu/ρb)(uf/U0)2/bracketrightbig are small for very lean ﬂames, which explains the similarity of the fresh gas ﬂow and the shape of the ﬂame wi th those of a bubble. Buoyancy dominates the upward motion of the ﬂame fro nt and the ﬂow of the fresh gas. The residence time of the hot gas in the low velocity region ma y be very large. This increases radiation losses, which cool down the slowly moving gas and may Upward propagation of very lean methane-air ﬂames in vertical tubes 1 1T/Tu v/UL x/δL 0 0 0.0002 1 0.0004 2 0.0006 3 0.0008 4 0.001 5 -8-6-4-2 0 2 4 6 8 0 0.001 0.002 -3-2-1 0 1 2 3 x/δLMass fractions(a) ωiMCH4δL//parenleftBig ρuULYCH4u/parenrightBig(b) Fig. 4(a): Proﬁles of temperature (red, right-hand scale) and mass fractions o f CH4(divided by 30, black), CO (divided by 15, dark blue), OH (green), and H 2(multiplied by 10, light blue) computed as functions of downward distance using the velocity proﬁles (dashed blue, right- hand scale) measured for φ≈0.53 and 0.54, increasing from bottom to top or as indicated by the arrows. ( b): Rates of steps 1 (red), 2 (green), 3 (dark blue), and 4 (light blue) of the four step reduced scheme. The values of the equivalence ratio determined from the semia nalytical model areφmodel= 0.509 and 0.519, respectively. Dotted red curves in ( a) are the temperature proﬁles computed with the single step kinetics. Temperatures are scaled with Tu= 300 K, velocities with the reference ﬂame velocity UL= 4.63 cm/s, distance with the reference ﬂame thickness δL=αu/UL= 4.32×10−2cm, with αu= 0.2 cm2/s, and reaction rates with ρu(YCH4u/MCH4)UL/δL. The origin of xis at the point where θdeﬁned in section 2.2 equals 0.8. have an inﬂuence on the ﬂame, even if the concentrations of ra diating species (mainly CO 2and H 2O) are very small; see below. Figure 4(a) shows temperature (red) and species concentrat ion proﬁles com- puted with the simpliﬁed model (1)–(14), together with the m easured velocity proﬁles used in these computations (dashed blue). Results a re shown for two val- ues of the equivalence ratio, φ≈0.53 and 0.54, the lower one being close to the minimum for which a ﬂame front can propagate in the tube for a w all tempera- ture of 26oC. The values of the equivalence ratio determined by the model are φmodel= 0.509 and 0.519, respectively, which are somewhat smaller tha n the real values and give an idea of the error of the model. For the same v alues of the equiv- 12 V. Muntean, F. J. Higuera alence ratio, Fig. 4(b) shows the rates (5)–(8) of the four-s tep reduced scheme as functions of the distance along the axis of the tube. Step 2 ( green) represents radicals production through the hydrogen-oxygen chain, while step 3 (dark blue) represents radicals recombination. Radicals are also cons umed to attack the fuel (step 1, red). The sum of the rates of steps 1 and 3 nearly coinc ides with the rate of step 2, implying that OH is not far from steady state. The rates of all three steps decrease when the ﬂame temperature is decreased by decreasi ng the equivalence ratio, but the rate of step 2, which has a large activation ene rgy, decreases the most and the rate of the three-body step 3 decreases the least . Eﬀective combus- tion ceases to be possible when radicals removal stops the fu el attack step 1, which happens below a certain cross-over temperature; see Refs. [ 17,18]. The results in Fig. 4(b) show that this condition is approached when φis decreased, but the rate of step 1 remains larger than that of step 3 at the smallest val ues ofφattained. Moreover, additional computations carried out for the same velocity proﬁles with the single reaction scheme, for which no cross-over tempera ture exists, give tem- perature proﬁles (dotted red curves in Fig. 4(a)) very simil ar to those of the four step scheme, which casts some doubts on the kinetic extinction account. Due to radiation losses, the maximum computed temperature is lower than the adiabatic temperature of the planar ﬂame ( Teq, withTeq/Tu= 5.12 and 5.19 for the two cases shown in Fig. 4) by an amount of the order of the Fr ank-Kamenetskii temperature, T2 eq/Ta, which is 0 .42Tufor the lower value of φand the activation temperature of the single reaction scheme. Thus the ﬂame is c learly nearing extinc- tion (see, e.g. Williams [9]), but the weak dependence of the temperature proﬁle on the kinetic scheme suggests that shortage of radicals need s not be the factor that unleash extinction; it may rather occur when the fate of t he ﬂame is already decided. The strong eﬀect of the radiation may be surprising at ﬁrst si ght. Acting on a planar ﬂame, the radiation losses considered here would de crease the ﬁnal com- bustion temperature by an amount δT=O/bracketleftbig λbLb/(ρucpuf)2/bracketrightbig , where Lbis the radiation loss (14) evaluated in the burnt gas (at T=Tb,YH2O= 9YCH4u/4, YCO2= 11YCH4u/4) andufis the ﬂame velocity; see Williams [9]. With ufof the order of the adiabatic ﬂame velocity and Tb≈Teq, this gives δT/(Tb−Tu) = O(2.5×10−2), which is too small to extinguish the ﬂame. The enhanced eﬀec t of radiation losses for near limit ﬂames is connected to the low velocity region that appears downstream of these ﬂames, as ﬁrst noted by Shoshin a nd Jarosinski [11] and Shoshin et al. [12]. Inspection of the convection, condu ction and radiation terms of the energy equation (10) downstream of the ﬂame [24] s hows that a convection-radiation balance analogous to that of a planar ﬂame is approached for values of φwell above the ﬂammability limit, but this balance changes t o a conduction-radiation balance near the ﬂammability limit b ecause convection be- comes small in the low velocity region. In these conditions, the decrease of the ﬁnal combustion temperature is given by the modiﬁed estimat e δT=O/parenleftbiggLbℓ ρucpuf/parenrightbigg , (15) whereℓis the characteristic size of the low velocity region, of the order of the radius of the tube. This modiﬁed estimate follows from an ent halpy balance across the ﬂame, ρuufcpδT=qb, where qbis the heat conduction ﬂux from the ﬂame to the burnt gas, which is of order Lbℓwhen conduction and radiation matter in Upward propagation of very lean methane-air ﬂames in vertical tubes 1 3 the low velocity region. This δTis larger than the estimate quoted above for a planar ﬂame by a factor ℓ/δb, whereδb=λb/ρucpufis the ﬂame thickness. This factor can easily be of the order of 20 or higher, which makes δTlarger than the Frank-Kamenetskii temperature and accounts for the results i n Fig. 4. Table 1 Mass fraction of methane and equivalence ratio at the ﬂammability l imit for various values of the N 2:O2volume ratio. N2:O20 0.9 1.0 2.9 3.8 5.3 6.2 YCH4u0.031 0.030 0.030 0.030 0.030 0.031 0.031 φ 0.13 0.14 0.23 0.44 0.54 0.72 0.82 Additional experiments have been done to further assess the e ﬀect of radiation losses. Experiments with various values of the fresh gas N 2:O2volume ratio show that the mass fraction YCH4uis nearly constant at the ﬂammability limit, while the equivalence ratio increases with the nitrogen dilution; see Table 1. This result can be accounted for noticing that ( a) a global enthalpy balance shows that the temperature diﬀerence Teq−Tuis proportional to YCH4uand independent of the N2:O2ratio if small changes of the speciﬁc heat and very small conc entrations of unreacted species in the equilibrium mixture are left out; and (b) the radiation losses (14) are proportional to the mass fractions of the rad iating species, whose values in the hot burnt gas are in turn proportional to YCH4u. Thus, if radiation losses are relevant, the diﬀerence Tb−Tu, whereTb=Teq−δTis the ﬁnal com- bustion temperature, and the whole structure of the ﬂow, sho uld depend only on YCH4uand not on the dilution. In particular, YCH4ushould have a constant value at the ﬂammability limit, in agreement with the experimental results. The ﬁnal combustion temperature Tbis also predicted to be constant at the ﬂammability limit, but we cannot measure this quantity in our experiments. Table 2 Mass fraction of methane at the ﬂammability limit for various va lues of the mass fraction of CO 2in the fresh gas. In all the cases ([N 2]+[CO 2])/[O2] = 3.76 in the fresh gas. YCO2u0.057 0.072 0.092 0.11 0.14 YCH4u0.03 0.031 0.032 0.032 0.033 Experiments have also been carried out replacing part of the N 2of the fresh gas with CO 2, liter per liter. Table 2 shows that the mass fraction of meth ane at the ﬂammability limit increases with the concentration of CO 2in the fresh gas. A line of slopes= dYCH4u/dYCO2u≈0.035 can be ﬁtted to the data. These results reﬂect the competition of two opposing factors. On the one hand, rad iation losses are increased by adding CO 2, both directly, due to the presence of this species already in the fresh gas, and indirectly, due to the additional radia ting species produced by the combustion of the additional CH 4. On the other hand, the additional CH 4 strengthens the ﬂame. The rates of decrease of the ﬁnal combu stion temperature due to enhanced losses and of increase due to the increased me thane concentration can be estimated and compared. 14 V. Muntean, F. J. Higuera 0.03 0.05 0.07 0.090.540.560.580.60.62 XCO 20.03 0.05 0.07 0.09145015501650 Ta−δT Fig. 5Measured equivalence ratio (blue, left-hand scale) and estimated ﬂame temper ature at the ﬂammability limit (black, right-hand scale) as functions of the mole fraction of CO 2added to the fresh gas. The red curve gives the adiabatic ﬂame temperature. –The sensitivity of the radiation losses in (14) to the added CO 2can be approxi- mated by L′ b=/parenleftbig ∂Lb/∂YCO2/parenrightbig [1+(44/16)s]+/parenleftbig ∂Lb/∂YH2O/parenrightbig (2×18/16)s, where the factors proportional to saccount for the additional mass of CO 2and H 2O present in the burnt gas due to the combustion of the addition al CH4. Carrying this approximation to the estimate (15) of the decrease of the ﬁnal combus- tion temperature due to radiation losses, we ﬁnd δT′= d(δT)/dYCO2u= O/parenleftbig L′ bℓ/ρucpuf/parenrightbig . –The rate of variation of the adiabatic ﬂame temperature due t o the variation ofYCH4uthat accompanies the variation of YCO2ufor the limit ﬂame is T′ eq= dTeq/dYCO2u≈sq/cp. Using the experimental value of sand replacing ℓandufin the estimate of δT′ with the radius of the tube and the adiabatic ﬂame velocity co mputed with the Cantera software, the ratio δT′/T′ eqcomes out of order unity for near limit ﬂames. This shows the important role played by radiation losses. Wh en the losses are increased by increasing YCO2u, the strength of the ﬂame must be increased by increasing YCH4uin order to oﬀset these losses. Figure 5 shows the values at the ﬂammability limit of the equiv alence ratio (blue curve), the adiabatic ﬂame temperature ( Teq, red curve), and an estimated ﬁnal combustion temperature (black curve). This temperatu re is obtained by sub- tracting from Teqthe value δTestimated in (15) [with L(Teq) evaluated from (14) at the equilibrium composition of the mixture] multiplie d by an arbitrary factor. The estimated ﬁnal combustion temperature is nearl y independent of the concentration of CO 2for a certain value of this factor. 4 Conclusions The upward propagation of a lean methane-air ﬂame front in a v ertical tube has been investigated using photographic records and PIV to mea sure the shape and velocity of the ﬂame front and the velocity of the gas around t he front. Upward propagation of very lean methane-air ﬂames in vertical tubes 1 5 A simple semianalytical model of the temperature and species mass fractions along the axis of the tube has been proposed which uses the meas ured velocity of the gas. This model gives reasonable predictions of the ﬂamm ability limit, in agree- ment with our experimental data, and suggests that extinction at the ﬂammability limit may be fairly independent of kinetic details. Series of experiments have been carried out in which the N 2:O2volume ratio of the fresh gas is varied or part of the N 2is replaced by CO 2. The mass fraction of methane at the ﬂammability limit is found to be nearly inde pendent of the N2:O2ratio and to increase with the concentration of CO 2in the fresh gas, which supports the view that radiation losses are important at the ﬂ ammability limit measured with the standard ﬂammability tube. Acknowledgements This work was supported by the Spanish MINECO through projects CSD2010-00010 and DPI2013-47372-C02-02.