asymptotic structure of counterflow diffusion flames_linan

7/30/2019 Asymptotic Structure of Counterflow Diffusion Flames_Linan

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Acta Astronautica. Vol. 1, pp. 1007-1039. Pergamon Press 1974. Printed in the U.S.A.

The asymptotic structure of counterflow diffusionflames for large activation energiesA M A B L E L I N A N

Instituto Nacional de Tecnica A eroespacial and E scuela Tecnica Superior de IngenierosAeronauticos, M adrid, Spain{Received 7 September 1973)

AbstractThe structure of steady state diffusion flames is investigated by analyzing the mixing andchemical reaction of two opposed jets of fuel and oxidizer as a particular examp le. An Arrhenius one-step irreversible reaction has been considered in the realistic limit of large activation energies. Theentire range of Damkohler numbers, or ratio of characteristic diffusion and chemical times, has beencovered. When the resulting maximum temperature is plotted in terms of the Damkohler number(which is inversely proportional to the flow velocity) the characteristic S curve emerges from theanalysis, with segments from the curve resulting from:

(a) A nearly frozen ignition regime where the temperature and concentrations deviations from itsfrozen flow values are small. The lower branch and bend of the S curve is covered by this regime.(b) A partial burning regime, where both reactants cro ss the reaction zone toward regions of frozenflow. This regime is unstable.(c) A p remixedflameregime where only one of the rea ctants leaks through the reaction zone, whichthen separates a region of frozen flow from a region of near-equilibrium.(d) A near-equilibrium diffusion controlled regime, covering the upper branch of the S curve , with athin reaction zone separating two regions of equilibrium flow.

Analytical expressions are obtained, in particular, for the ignition and extinction conditions.1. IntroductionTwo basic classes of non premixed combustion problems can be defined as"initial-value p rob lem s", described either by ordinary differential equations withboundary conditions at only one point or by parabolic partial differential equations, and "boundary-value problems", described by ordinary differential equations with two bo undary points or by elliptic partial differential equation s. From aphysical point of view, solutions to initial-value problems are unique, but multiplesolutions to boundary-value problems may exist corresponding to drasticallydifferent combustion regimes, this being especially true for large activationenergies.Stagnation-point counterflow of fuel and oxidizer [1-7], comb ustion of a spherical fuel droplet in an oxidizing atmosphere [8-12], planar one dimensional inter-diffusion and reaction of fuel and oxidizer [14-16], are among the boundary-valueprob lems that hav e been studied in efforts to clarify diffusion-flame struc ture . Weshall carry out in this paper an asymptotic an alysis for large activation energies ofdiffusion-flame structure in problems of th e bound ary-value type . Since the qualitative aspects of flame structure are independent of the specific choice of problems within th e bound ary-value class, we hav e selected the counter-flow diffusionflame as a model.

1007


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In unpremixed combustion, the maximum temperature in the flow field depends on a Damkohler number, defined as the ratio of a diffusion time to areaction time evaluated at a boundary temperature. For low activation energiesthe dependence is represented by a monotonic function which goes from thehighest boundary temperature at zero Damkohler number to an adiabatic flametemperature at infinite Damkohler number. For high activation energies andadiabatic flame temperatures sufficiently higher than the boundary temperatures,the curve develops an S shap e, with the ignition conditions clearly defined as theposition of the low-temperature vertical tangent and the extinction conditions bythe high tem peratu re vertical tangent. That only the portion of the curve betweenthe vertical tangents is dynamically unstable has been demonstrated by transientstability analysis, under suitably restricted conditions [16],The curve giving the maximum temperature vs the Damkohler number hasbeen generated by somewhat laborious numerical solutions of the steady stateequations [3-8 ,15,16] and by approximate m ethods of an ad hoc nature [1,8-10].Asym ptotic methods w ere first used in the form of boundary layer type analysis ofnon-equilibrium effects in the limit of large Dam kohler num bers [2,14]. La ter singular perturbation metho ds w ere used [3,11,12 ] to refine the asym ptotic solutionby retaining higher order term s in an expansion for large Damkohler numb ers. A;regular expan sion for low Damkohler nu mbers w as also carried out [3] for thestagnation point diffusion flame. Asymptotic methods that treat the Damkohlernumber as the large or small parameter are incapable of defining ignition orextinction conditions so that wide use has been made of numerical methods forthis purpose.Adoption of the activation energy, or more specifically the ratio of activationtemperature to the characteristic temperature of the system, as the large parameter in the present work produces analytical results for the entire range of theDamkohler number, readily exhibits the various parametric dependences, accurately covers conditions of greatest practical interest and generally provides amuch clearer physical understanding of the process involved.In Section 2 we formulate the problem and identify the limiting solutions andcombustion regimes. In Sections 3 to 6 we consider in detail the ignition, partialburning, premixed flame and diffusion flame regimes. In Section 7 we summarize the results and compare with other numerical results.Initial value problems in combustion have been analyzed for large activationenergies by Lifian and Crespo[25].2. Formulation

We 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 Arrhen ius reaction will be considered, which is first orde r with respe ct to bothfuel and oxidizer.Even though the asym ptotic analysis may be performed easily for more complex fluid dynamic systems, we shall assume, for simplicity in the presentation,that the two streams have equal velocity and that the density, specific heat, and


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transport coefficients are constant. We shall use Fick's law, with equal dif-fusivities of mass and heat, to calculate the diffusion velocities. The Schvab-Zeldovich procedure then leads to the following equation for describing thetemperature field (see Ref. [3] or [4] for the details of the derivation):^ ? + z | j = - D y 0 y p e x p ( - T a /T), (1)

with the boundary conditionsT=TX at z-> and T = T-~ at z -* -o c (2)

Here T is the temperature made nondimensional with the characteristic temperature QICpYF- obtained from the heat relea se per unit mass of fuel Q, the specificheat cp and the fuel mass fraction in the fuel stream. Th e distance no rmal to themixing layer z, measured from the plane of the stagnation point has been madenondimensional with the characteristic mixing length VD t f/A where D if is thediffusion coefficient and A is the stagnation point velocity gradient, or ratio between velocity and distance normal to the mixing layer. The fuel stream approaches from z = oo, and the oxidizer stream from z = + oo where the oxidizermass fraction I S l o o c . The Damkohler number D = ZvYF-JA is based on thefrequency factor Z of the reaction ra te and the characte ristic flow time A "1 of thevelocity field, with the stoichiometric mass ratio v of oxidizer to fuel appearing asa factor. Ta is the nondimensional activation energy. The factors y0 and yFappearing in (1) are the oxidizer and fuel mass fractions divided by vYF-x andYF_*, respectively. They are given by the relations

yQ=a(\-x)+T f-T (3)yP=x + T f-T (4)T f=T- j3x

x = ( l / 2 ) e i / c ( z / V 2 ) ( 5 )T f is the nondimensional frozen flow temperature with j8 = (T T-o) which weassum e, without loss of generality, to be larger than zero . The normalized oxidizermass fraction at the oxidizer stream is a = Y0=c/vYF-=o.Equations (1) and (2) when written in terms of x take the following formj^ = - 2TT exp (z2)D y0yF exp ( - TJT) (6)

where the factor 27r exp (z2) = (dzldx)2 should be written in terms of x by using(5); its dependence on x results from the convective term in Eq. (1).The boundary conditions are

T = TX at J C = 0 and T=Tx-(3 at x = 1 (7)Then, the normalized fuel and oxidizer con centrations are , respectively, 1 and 0 atx = 1, and 0 and a at x = 0.

withwhere


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Our pu rpose is to obtain the asymp totic solution of (6) and (7) for large values ofTa, with a, f$ , T of order unity and the Dam kohler num ber D ranging from zero toinfinity.If in Eq. (6) we take the limit Ta -* and consider D to be a function ofTa, the following limiting equations are obtained depending on the resulting localvalue of T and our choice of D(T a):(a) Frozen flow with

d2Tjdx2 = 0 (8)Therefore T is a linear function of x.

(b) Equilibrium flow with zero oxidizer concentrationy0 = 0- T = a ( l - x ) + Tco-j3x (9)

(c) Equilibrium flow with zero fuel concentrationyF = O - r = jc + T -0 ; c (10)Th e four different possible regimes that w e shall encoun ter for large Ta showthe ex istence within the flow field of two reg ions, where th e tem peratu re is givenin the first approximation by Eqs. (8) to (10), separated by a thin reaction zone.The reaction zone is infinitely thin in the limit Ta -* oo.fThe flame temperature is a possibly multivalued function of the Damkohlernumber, so that ignition-extinction condition will be exhibited. When the flame

temp erature varies from the boundary temperature Too to th e adiabaticflame emperature we may encounter a nearly frozen ignition regime, a partial burning regime, a pre-m ixed flame regime and finally a diffusion contro lled o r diffusion flam eregime. W e shall describe below these combustion regimes. For the temperaturedistributions, T as a function of x, corresponding to these po ssible regimes, thereader should refer to Fig. 1 for the ca ses w here 0 > 1/2, to Fig. 2 for /3 < 1/2 anda + 20 > 1, and to Fig. 3 fo r /3 < 1/2 and a + 2)3 < 1. The temperature distributioncorresponding to the diffusion flame regime, for example, is given by two segments of the lines y0 = 0 and yF = 0. Th e tem perature distributions have beenshown for frozen flow (F.F.), and for two examples of the partial burning (P.B.)and prem ixed flame (P .F.) regimes. The thick dashed line in these figures gives thepossible flame temperature and the flame location for all Damkohler numberranging from zero to infinity.

1. Frozen flow Ignition regime. In this regime the chemical reaction isfrozen in the first approximation and the temperature is given byT~T f = Too- /3JC (11)

which satisfies the boundary conditions (7). The ox idizer and fuel concen trationstLibby and Economos[24] introduced a distributed flame-zone model of combustion in a non-premixed system, in which the reaction was treated as being frozen below an artificially introducedtemperature. This model is not applicable to the present kinetic scheme.


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The asymp totic structure of counterflow diffusion flames 1011

| 2 3 > 1 |-TBD+OC

X 1Fig. 1. Asymptotic temperature and concentration distributions for the frozen flow(F.F.), premixed flame (P.F.), and diffusion controlled (D.F.) regimes in terms of thestrained coordinate x. N o partial burning regime exists in this cas e, wh ere Te < T.

Fig. 2. Asym ptotic temperature and concentration distributions for the frozen flow, partial burning (P.B.), premixed flame, and diffusion controlled regime for the case 2/3 < 1and a + 2j8 > 1, when theflame ies always in the lean side of th e mixing layer.


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2 B < 1oc + 2 0 < 1

Fig. 3. Asymptotic temperature and concentration distributions for the case when a +2/3 < 1. In the prem ixed flame regime the flame ies on the rich side of the mixing layer.are then given by

y0/ = a(\ - x) and yF/ = x (12)To obtain the temperature in the second approximation, in an expansion forlarge TJITa, chemical reaction effects must be retained. For a sufficiently largeDamkohler number the temperature rise above the frozen flow value is of orderTJlTa, so that the nonlinear effects associated with the Arrhenius exponent areimportant. These effects are responsible for a rapid increase in the productconcentration for large Dam kohler num bers w hen j8 > 1, and for the existence ,when 0 < 1 of two solutions if D is lower than an "ignition Dam kohler n um ber"and no solutions above this value. Hence the lower bend of the S curve isdescribed by th e ignition regime an alysis. The condition /3 < 1 is equivalent to thestatement that th e boundary temp erature T is lower than the adiabatic flametemperature Te = Tx+ (1 - 0 )a / ( l + a).2. Partial burning regime. In this regime two frozen flow regions are separated by a thin, infinitely thin in the limit Ta ~*


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Th e asymp totic structure of counterfiow diffusion flam es 1013

The flame is located at x = xh where the temperature is Tb and the fuel andoxidizer concentration yFb and yob as given by (3) and (4).As we shall show below, the matching conditions with the asym ptotic solutiondescribing the temperature field within the thin reaction zone (a) indicate that thetemperature distribution, obtained from (13) and (14), in the outer frozen flowregions should d ecrea se at the same rate on both sides of th e flame and (b) providea relation between Th and the Damkohler number D. From the equality of theslopes of the temperature distribution on both sides of the flame we deduce

x b = { 2 + ^ / ( T b - T , ) } - 1 (15)This regime is only po ssible for j3 < 1/2. The flame temp eratu re Tb is lowerthan T j8 + 1/2 when a + 2j8 > 1, otherwise yFb would become negative. When

a +2/3 < 1, Th is limited by T+ a/2 so as to have yob > 0 .3. Premixed flame regime. In this regime a frozen flow region, and a equilibrium region a re sepa rated by a thin reaction region with leakage of only one of thereactants through the flame. The structure of the reaction zone is the same as thestructure of the reaction zone of a premixed flame with heat loss toward theequilibrium side of the flame. Two possibilities arise depending on the value ofa + 2 j 3 .For a + 2/3 > 1, we find equilibrium flow with yF = 0 for x < xp

T = T + ( l - / 3 ) x (16)and frozen flow for x > xp, withT = T - - j3 + (T p - T .+ 0)(1 - x) /( l - x p ) (17)Here Tp is the value of T at x = xp as obtained from (16). At the flameyo = yop = a - (a + \)x p.For a + 2(3 < 1, equilibrium, with y0 = 0, occurs for x > xp, with T given by

T = T + a - ( a + /3)x (18)and frozen flow for x < xp, with

T = T . + (T P -T.)(x/Xp) (19)In this case at x = xp, yP = yFp = - a + (a + \)xp.The two alternatives result from the fact that matching with an inner reactionzone solution, with com plete burning of one reac tant in the first approxim ation, ispossible only if the tem peratu re decays tow ard the equilibrium side of the flame ata slower rate than toward the frozen flow side. The matching conditions, inaddition, will provide a relation between Tp and D.

4. Diffusion flame or near-equilibrium regime. For sufficiently highDamkohler numbers a diffusion controlled regime exists in which equilibrium isreached , to the first approximation, on both sides of a thin reaction zon e. Then,x < x : Y F = O - * T = T , . + ( l - 0 ) x (20)j c > x e : Yo = 0 ^ T = T = c + a - ( a + ^ )x . (21)


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At the flame the temperature takes the adiabatic flame valueT . = T . + ( l - j 3 )a / ( l + a ) (22)

and the flame is located at xe,xe = al(a + \) (23)

We shall see below, when analyzing the structure of the thin reaction zone,that for /3 < 1 and sufficiently high activation energ ies multiple solutions m ay existfor D amkohler numb ers above a minimum "extinction D " and no solution for Dbelow this value.There are cases in which the extinction conditions do not occur under thisregime but under the previous premixed flame regime. This is the case, forexample, if (1 /3) is small compared with unity.In all these regimes the asymptotic flame structure is determined, in the firstapproximation, independently of chemical kinetics, with the exception of thepartial burning and premixed flame regimes in which an unknown flame temperature, Tb or Tp, appears which will be a function of the Damkohler number. Therelation betw een flame tem perature and Dam kohler number will be o btained, sequentially for the different regimes in the following sections, from the matchingconditions between the asy mp totic expansions for the tem perature in the reactionzone and in the outer regions.3. Nearly frozen, ignition, regime

As indicated before, in this regime the Damkohler number is such that thetemperature rise, above the frozen flow value, due to the chemical productionterm, is of order TJlTa, which is small com pared with TV, but large enough so asto force us to retain in the equations the nonlinear effects associated with theArrhen ius exponen t. Th ese effects are responsible for the ignition chara cteristics.We shall use e = Tj/Ta as the small parameter, and begin by analysing thecase where the tem perature difference j3 between the two streams is of order e.We 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. Wechoose D as the depende nt variable because D is a single valued function of thattemperature increment while the inverse function may be multivalued.We shall introduce in Eq. (6) the expansionsT = T o O + e ( 0 1 - / 3 , x ) + e 2 a 2 + . (24 )

Da(TaITj) exp ( - TJToo) = A0 + eA t + (25)whe re 0 i ( x ) , . . . , A 0 ,..., which we anticipate to be of order unity, are to bedetermined in terms of 0 t(l/2), with 02(l/2) = 03(l/2) = 0, or chosen for convenience; we wro te /3i = /3/e. Th e following equation is obtained for 0i

d26-r4 = - 2TT exp (z2)Ao(l - x)x exp (0 , - /3,x) (26)to be solved with th e bo und ary conditions 0i(O) = 0i(l) = 0.


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The value of 0r(l/2) obtained from Eq . (26) depend s on the reduced Dam kohlernumber A0, or vice versa. Equation (26) may also be obtained from Eq. (6) byneglecting the reactant consumption and linearizing the Arrhenius exponentaround the higher boundary temperature in the manner of Frank-Kamenetskii[17]. There are no solutions of this equation for A0 larger than acritical "ignition " value Aw which is a function of $u and there are two solutionsfor Ao < AOJ. Of course if we specify 0i(l/2) we then obtain a unique A0 lower thanAoi.Figure 4 shows $(x) for /3i 0 and several values of A0 as obtained from a

Ao= D(T a /U)exp( -Ta /T.)Fig. 4. Temperature distribution in the nearly frozen regime, for p - 0 and several values of the reduced Damkohler number A0. The lower part of the figure shows themaximum temperature as a double-valued function of the Damkohler number for Ao


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numerical solution. Also shown in Fig. 4 is #t(l/2), the maximum temperature inthis case, as a function of A0; the lower bend of the S curve is thus reproduced.A first approximation for the Damkohler number of ignition, obtained fromEq. (25), is given bye"'D,a exp {-TJTJ = A 0,(j3/e). (27)

The reduced Damkohler number of ignition A 0i(j3,), represented in Fig. 5, increases with increasing /3 t.18

16

H

12

10

8

6

2

0 1 2 3 4 5 6 7 8PieFig. 5. Th e reduced Dam kohler number of ignition A0, in terms of /3/e.

For large values of j8 t, the chemical reaction, is limited to a region near the ho tboundary such that (3xx is of order unity; for larger values of x the Arrhenius factor becomes exponentially small. The analysis of the nearly frozen regime forlarge values of j3i, or /3 of order unity, is left for the Appendix A. There we obtainthe following asymptotic expression for the Ignition Damkohler numberD ta exp (-TJT cc) = 2e~2(l - j3)-2F(/3) In {/32(1 - jS)2Ta2/87rT.4}, (28)

where F(j8) is very accurately correlated by F(0) = 2/3 - /32.Multiple solutions, and thereby clearly defined ignition conditions, are obtained only for 0 < 1.For close to on e, in the ignition conditions the reaction zone struc ture is ofthe sam e form as that of th e premixed flame regime, and then the results obtainedin Section 5 may also be used to determine the ignition conditions. For small


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values of j8, although large (31, the Dam kohler num ber of ignition can be determinedby both Eqs. (27) and (28).4. Partial burning regime

In this regime, which is only possible for /3 < 1/2, two regions of frozen floware separated by a thin reaction region. If the activation temperature is largecompared with the local temperature a very small change in this temperaturebelow the maximum is necessary to freeze the chemical reaction. Both reactantsleak through the reaction zone w here they u ndergo a very small relative change inconcentration.The Damkohler number is large enough to make the reaction time at themaximum temperature of the same order as the diffusion time through the reaction zone . In the frozen flow regions the tem pera ture, w hich, for large Ta, is givenin the first approximation by E qs . (13) and (14), being sm aller than the maximumtemperature, Tb in the first approximation, produces a chemical reaction timeexponentially large in TalT compared with the diffusion time. Only at both outeredges of th e mixing layer will, again, the large reaction tim e be comparable with anappropriately large residence time in thick zon es, where th e chemical reaction willgo to completion without a significant increase in temperature. The existence ofthese thick reaction zones at the outer edges does not affect, to any algebraicorder, the solution in the frozen flow regions or in the interior reaction zones andtherefore we shall not analyze it here.Peskin and Wise presented in Ref. [9] a method for the analysis of non-equilibrium diffusion flames in which they assume the reaction zone to be infinitely thin allowing both reactants to leak through the flame. However theychose the flame location and the amount of reactants burning in the thin flameartificially, instead of basing this choice on the matching conditions with anasymptotic reaction zone solution for large activation energies.We shall use e = Tb2ITa, as the small parameter, and pose our problem as tofind the Dam kohler number and the asymptotic solution for small e of Eqs. (6) and(7), when the temperature in the frozen region to the right of the reaction zone,(JC ~xb)le > 1, is given to all algebraic orders in e, by Eq, (14), with xb given interms of Tb by Eq. (15).In the frozen flow region tow ard th e oxidizer side, (x b - x)fe > 1, the chemicalreaction will also be frozen to all algebraic orders in e. Ho we ver, the tem peraturewill be given by Eq . (13) only in the first approximation, a s w e shall see below. Weuse the following outer expansion for this region,

T=T + (T b- TJ(xlxb) + eBxx + ezB2x + (29)which reflects the fact, that th e chemical reac tion is frozen to all algebraic ordersin e.For the temperature within the inner reaction zone we use the expansion

T=Tb+ e


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We also expand D as followsD = A d + bxe +fc2e2+ ) (32)

whe re A will be the Dam kohler number which makes matching possible, to thefirst approximation, with the outer solutions (13) and (14). If (30) and (32) areintroduced in Eq. (6), previously written in terms of we obtain the followingequation for 4>i, free from con vec tive effects,

d ^ A t f ^ - A o e x p * . ( 3 3 )whereAo = (T b2/T a)27r exp (zb2)Ayoby exp ( - Ta/Tb)yob=a + T~Tb~(a + /3){2 + 0/(T - T.)}"1y = r . - Tb + (1 - 0){2 + fiKTb - r - ) } -are the oxidizer an d fuel co ncen tration in the reaction zo ne in the first approximation; zb is the value of z, as given by Eq. (5), at x xb. From Eq. (33) we obtain(d&JdO 2 = 2A0(exp m - exp ,) (34)

where $ m is the maximum value of ,. From Eq. (34) we obtain2ln(t{l + VT =T7?} ) = (2A0 exp


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relation must be ob tained from the m atching conditions with the three term exp ansion of the inner so lution. Th e details are left for th e Ap pendix B whe re we o btainB , = 2{(a + j3)/y0b + (j8 - l)/y r a - A-nzb exp (zb

2)} (40)Equations (36) to (40) may be used to determine, in terms of Tb for example,the maximum temperature Tb + em, and its location xb + em, in the second approximation. We may simultaneously obtain Bu which gives the temperature inthe frozen flow region x < xp, and D b, the first approximation for the Damkohlernumber. Explicitly,

D byobyn> exp ( - TJTb) = (T a/Tr)(Tb - T)2(T bxby2 exp ( - zb2 + O m) (41)where$ m = - In 4 + xb {(a + p)/y0b + (j8 - l)/y Fb - 47rzb exp zb

2} (42)Th e range of flame temp erature s in this partial burning regime is limited by th econdition that yob and y ^ must be larger than z ero . Th e upp er limiting values of Tbare, therefore, Ti = T+ a/ 2 for a + 2)3 < 1 when y0b becomes zero, and T2 =Tc-jS + 1/2 for the case a +2/3 > 1 when y^, becomes zero.The flame temperature, Tb, vs Damkohler number, D b, resulting from (41) willhave an S shaped form, exhibiting thereby, ignition extinction characteristics.However, of the three branches of the Tb(Db) curve, the lower, the ignitionbranch, will correspond to small values of Tb T that are not large compared

with TJlTa; so that the analysis given in this section is no t valid for this bran ch,because th e thickness of the reaction zone becom es of orde r unity if /S TJlTa,or of order xb if /3 > TJITa. Thus the analysis given in the previous section mustbe used to describe the ignition branch.The middle branch, where Tb increases with decreasing D b, will very likelyturn out to be unstable in a stability analysis similar to that presented by Kirkbyand Schmitz[15,16].A third upper branch of the curve Tb(Db) exists because D b goes to infinitywhen y0b or yFb go to ze ro; in this case, the expansion for the reaction zone willfail, and the p remixed flame analysis of the following section should be used . Onlyin the particular c ase a + 2/3 = 1, for which y0b an d y** go to zero simultaneously,will the diffusion controlled regime appea r sequentially to th e partial burning regime when Tb approaches Te. The extinction conditions cannot be determinedfrom the analysis of this regime.5. Premixed flame regime

While in the previous regime leakage of both reactants through the flame occurred, in this premixed flame regime the flame acts, in the first approximation forlarge Ta, as a comp lete sink for one of th e reacta nts, the fuel if a + 2/3 > 1, and theoxidizer if a + 2/3 < 1. In this approxim ation th e temp eratu re is given by Eq s. (16)and (17) or (18) and (19), the re sults co rresponding to frozen flow on o ne side ofthe flame and equilibrium flow on the other.If /3 > 1, the tem peratu re in the reac tion zon e, which is close to Tp, will belower than th e temperatu re in the equilibrium region, which ranges from Tp to T,;


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therefore the flow in this region will be in equilibrium to all algebraic orders inTa~\ This is not the case if 3 < 1, because then the temperature reaches itsmaximum value within the reaction zone, and the flow outside will be chemicallyfrozen, to all algebraic orders in Ta'1; the fuel and oxidizer after crossing thereac tion z one will coexist in frozen flow. If, in the first app roxim ation, equilibriumflow exists on one side of the flame it is bec ause, to this approxima tion, com pletecombustion of one of the reactants occurred in the reaction zone before thetemperature decreased, toward the near-equilibrium side, enough to freeze thechemical reaction. However, leakage of this reactant through the flame will occurin all higher order approximations.As mentioned before, the asymptotic solution for large Ta of E q. (6) is given inthis regime by Eqs. (16) to (19) independently of chemical kinetics if Tp isspecified. To obtain higher order approximation to the flame structure and therelation between Tp and the Damkohler number we shall proceed as in theprevious regime. The detailed analysis that follows corresponds to the case a +23 < 1 in which no leakage of oxidizer through the flame occurs in the firstapproximation. The analysis is completely analogous, and the results may bedirectly written, for the case a +23 > 1.The problem we po se is to find the Dam kohler number and the asymptotic solution of Eqs . (6) and (7), for low values of e = Tp2/Ta, when the temp erature in thefrozen flow region, (x p - x)/e > 1, to the left of the reaction zon e is given by E q.(19) with JC P and Tp related by

xp=(Tx + a~Tp)/(a + B) (43)For the temperature distribution in the near-equilibrium region, (x - xp)/e > 1,to the right of the reaction zone, we use the expansion

T - T - B +(a + 3)(1 - x ) + eP,(l - x ) + e2P 2(l - x ) + , (44)in which we hypothesize that chemical equilibrium, with y0 0, occu rs in the firstapprox imation. H ow eve r, the chem ical reaction will be frozen to all algebraic orders in e, because the temperature in this region will be lower than Tp and theDamkohler number is not high enough to offset the effect of the Arrhenius exponent. Therefore leakage of the oxidizer through the flame occurs in the secondapproximation, proportional to

- (dyjdxh.t = - ePy - e2P2+ (45)For the analysis of the reaction zone we use the inner variable

i7 = a (x - xp )/x pe - p0/m (46)and the expansionT = Tp -e{m


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anticipate to be of order unity, is the first term of an expansionP =P o + Pi + - (49)

of p = In {4TT exp (zP2)yFp(jc pTp2/T0)2D exp (-T/T p)} (50)The first approximation for the fuel concentration yFp appearing in Eq. (50) is

yFp = (1 + a)x p - a (51)and zp is the value of z corresponding to xp.If we introduce the expansions (47) and (49) into Eq. (6) previously written interm s of 17, we obtain the equ ation, free from con vec tive effects, for y,,2 d 2 y , / dT 7 2 =y , e xp - ( y 1 + mi?) (52)Equation (52) is to be solved with the boundary conditions obtained from thematching conditions with the outer solutions given by Eq. (19) and (44). Wetherefore require,TJ -* - 00 : dyjdrj = - 1 (53)T J - + O O : dy,/di7=0 (54)

The choice of the stretching factor for the inner variable was made in order toobtain the normalized boundary condition (53). Similarly the choice of the translation point was made so as to obtain the factor 2 in the left hand side of Eq. (52).This is the only value for which Eq. (52) has solutions satisfying the boundaryconditions (53) and (54) in the limiting case m = 0.Figure 6 shows yi(i7), for different values of m, as obtained from a numericalintegration of Eq. (52). The results are also given for negative values of m,because in the analysis of the premixed flame regime for values of )3 larger thanone we encounter negative values of m. Notice that the exponential factorexp(- mrj) in Eq. (52) represents the effect of the heat loss, or gain, from thereaction zone toward the near-equilibrium side of the flame.Figure 7 shows the limiting values,

n = lim(y i +17), for 17 -00 (55)and yi- = limy i, for 17 -*oo (56)as a function of m.For small values of m, n = 1.344 and y, = 0. Both yloo and - n grow to infinitywhen m approaches 0.5, and there is no solution of Eq. (52), with the boundaryconditions Eqs. (53) and (54), for m > 0.5. The limiting value of m correspond toequal rate of decrease of the temperature on both sides of the reaction zone. Thetemperature must, therefore, decrease at a slower rate toward the near-equilibrium side than tow ard the frozen flow side , so that matching is possible between the inner diffusive-reactive solution of Eq. (52) and the first approximationfor the outer solutions.When m approaches 0.5, as shown in Appendix C, yi. and n grow towardinfinity as 1/(1-2m) and {(1 - 2 m ) " 1 + 21n(l - 2 m ) } respectively. For negative


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1022 A. LINAN

T i i i i i i r

Fig. 6. Internal structure of the reaction zone for the premixed flame regime. The non-dimensional reactant conce ntration, yi, is plotted a s a function of a nondimensional coordinate normal to the flame, rj, for sev eral values of the parameter m. The correspondingleakage of the reactan t, y lm , has also bee n indicated in the figure.values of m, yi,0. For large negative values of m, n behaves as {-ln(2m2)+2yB}lm, where yB is the Euler constant yE = 0.577; and the solution yi(r}) takesthe form y t = (2/m)KQ(t), where K0(t) is the modified Bessel function of orderzero and t = (V2/m ) exp ( - my 12).The matching conditions yield the anticipated boundary conditions (53) and(54), and in addition provide the relations

p 0 = - nm (57)andP 1 ( l - x p ) = -y , (58)

Equation (57) is the desired relation between the first approximation for theDamkohler number D p and the flame temperatu re. Equation (58) gives the am ountof oxidizer leaking through the flame, which therefore is of order e.We write Eq. (57) explicitly asD pyFp exp ( - TJTP) = (\/47r)(aTafx pTp2)2 exp ( - zp2-nm) (59)

Here nm is a function of m, and therefore of Tp, which has also been representedin Fig. 7. In the case a + 2/3 < 1 that we have considered m decreases from 1/2 to


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Th e asymptotic s tructure of counterflow diffusion flames 1023

Fig. 7. The parameters y, and n, defining the asymptotes of the relation y,(-rj), are hereshown as functions of m. No tice that y^ 0 for m < 0.(a + /3)/(a + 1), when Tp increases from its lower bound T+ a 12 to its upperbound Te.We shall now present the results corresponding to the case where a + 2/3 > 1,for which nearly com plete consump tion of the fuel tak es place in the thin reactionzone.In the first approxim ation, the flame temperatu re Tp, the flame location xp, andthe oxidizer concentration y0p are related by

JCP = ( T P - 7 ^ / ( 1 - / 3 ) ( 6 0 )yo P = a - (a + \)Xp (61)

For the analysis of the reaction zone we introduce the inner variablei7 = - ( 1 -p)(x- xp)lme - pQlm (62)


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A. LINAN

where m is now, given bym = l - ( r p - T + jS) (63)

and to define p, instead of Eq. (50), we usep = In {4TT exp (z p

2)(mTp

2/T a)

2(l - pr

2DyQp exp (~Ta/Tp)} (64)W e use Eq. (47) as the inner exp ansio n, and E q. (17) for th e temp erature distribution to the right of the reaction zone. For x 1/2, Tx and Teare the limiting values of Tp, so that l - /3 and (l - /3)/(l + a) are the corresponding limits of m ; we thu s find m to be n egative only if (3 > l, and zero of /3 = I.In the particular case fS = l, the flame temperature Tp is, in this regime, eq ualto To,, so that m = 0; then we should use xp as the independent variable. The

factor (l - p)lm appearing in Eqs. (62), (64) and (67) must be substituted by( l -xp)~ l; and in this case p0/m n - 1.344.It is clear that this premix ed flame analysis will not be valid for m close to itsupper limit 1/2, beca use then both point - - n and y, becom e infinite; leakage ofthe lean reactant through the flame increases, so that the equilibrium approximation on one side of th e flame canno t be used as the first order ap proxim ation. Th epartial burning analysis of the previous reaction should be used for m > 1/2.When Tp becomes close to Te the concentrations of both reactants in thereaction zone are small and have relative variations of order unity; then we mustuse the analysis of the near-equilibrium, diffusion flame regime that follows. Th eextinction conditions occur very often under the premixed flame regime.When j8 > 1/2, Tx is a bounding value of Tp, and the premixed flame analysisshould be replaced by the analysis of the nearly frozen regime if (T p - Tc=) becomes small. The ignition condition may also occur under the premixed flameregime if (l - /?) is small.6. Near equilibrium, diffusion flame, regime

Finally, a diffusion controlled regime exists for which the flow is eve ryw herenear equilibrium, so that in a first appro ximation the flame position an d tem perature distribution are determined by Eqs. (20) to (23) independently of chemicalkinetics. This corresponds to the Burke-Schumann[l8] and classical diffusionflame analysis [19].


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The asym ptotic structure of counterflow diffusion flam es 1025

We shall evaluate below, for large activation energies, the non equilibriumeffec ts in the range of Damkohler numbers for which the tempera ture devia t ionsfrom equilib rium a re of ord er e = Te2/Ta < 1. Th e nonlin ear effects associatedwith the Arrhenius expon ent a re responsib le for the ex is tence i f /S < 1 of extinctioncon ditio ns, which o cc ur in this regime if (1 - /3) is no t very small . By extinctionconditions we mean that multiple solutions of Eq. (6) exist for Damkohler number s abo ve an extinctio n value , and only a nea rly frozen solution exists for sm allervalues of D. Because the Damkohler number is a s ingle va lued funct ion of thedeviations of the temperature from equilibrium, and not necessarily vice versa,we shall po se our pro blem as to f ind D and the asymptotic solution of Eqs. (6) and(7), for a deviation, of order e , of the temperature from its equil ibrium value at agiven point, x = xe for example .We in troduce the outer expansion

T = T + (1 - j3)x - eA lx - e2A2x + (68)T = r 0 = + a - ( a + j 3 ) x - e B 1 ( l - x ) - e 2 B 2 ( l - * ) + ( 6 9 )

for large negative and posit ive values of (x xe)/e, respect ive ly . When wri t ingthes e expan sions w e assum e, in the f irst approx ima t ion , chemica l equi libr ium withyF = 0 or y0 = 0. W e, in addit ion, an ticipate th at in the oute r regions we ha ve , to allalge braic ord ers in e, froz en flow, or eq uilib rium flow if A; = J3j = 0.For the tempera ture d is t r ibut ion in the reac t ion zone we in t roduce the expansion T = Te- So-^CejB, + ey + e 2)8 2+ ), (70)w h e r e (3 U j8 2, , are fun ctions of th e in ner varia ble

= 8Qv\a + l)(x - xe)/2e. (7.1)In Eq. (70) the parameter

7 = 1 - 2(1 - j3)/(l + a) = 2(a + j3)/( l + a) - 1 (72)is such tha t (1 - 7) is tw o times th e ratio be tw ee n th e heat lost from the f lametoward the region of negative (x xe) and the total chemical heat release at thef lame. W hen 7 = 0 the hea t losses f rom the f lame tow ard both s ides a re equal :when 7 > 0 the tem pera ture dec rease s fas ter tow ard the reg ion x > xe than towardthe region x < xe. For 7 > 1 the f lame receives heat by heat conduction from theregion x xe < 0. The parameter 8 0 , which we anticipate to be of order unity, isthe f irst term of an expansion,

8 = 8o + edi + - , (73)of the reduced Damkohler number

8 = 8ir exp (z e2)(Te2IT af(a + \)~2D e x p ( - TJTe) (74)The length scale for the inner variable and the definit ion of the reducedDamkoh le r number 8 w ere ch ose n so as to obtain th e following equation for j8i,

d%ld2 = (75)


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1026 A. LINAN

where no convective effects appear, to be solved with the conditions,dpjd = l for -*> (76)dpt/d^-1 for -*~oo (77)

obtained from the matching conditions between the inner and outer expansions.The matching conditions do also provide the relationsA tx,8om = lim(p1 + 0* for -*-> (78)and Bi(l-x.)8oUi*)im(Pi-), for -*oo (79)

which enable us to calculate the distribution of reactants leaking through thereaction zone.Equa tions (75) to (77) describe in the first approximation the non equilibriumeffects, as dependent only on the redu ced Dam kohler numb er and y. The solutionsfor positive and negative y are trivially related, so we shall consider below y > 0 .Higher order approxim ations could be determined b y insuring, for exam ple, thatthe temperature obtained from the tw o term expansion at = 0 is not modified byhigher order effects. We shall not calculate them here.The retention in Eq. (75) of the exponential Arrhenius factor is essential forevaluating the non equilibrium effects in the near-extinction conditions. Eq. (75)with the boundary conditions (76) and (77) has tw o solutions, if | y | < 1, for 80above an extinction value 80B(y), and no solutions for 80 80B. Thu s, we find for y < 1 a minimum Damkohler num ber forthe existence of a near-equilibrium solution, and this may be used a s an extinctioncriterium. When y approaches 1 from below 8B, goes to zero. When y > 1 thesolution is unique.


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NEAR -EQUIL. DIFF. FLAME STRU CTUR E

Fig. 8. Th e internal struc ture of the reaction zone in the near-equilibrium, diffusion controlled, regime for the case 7 = 0. No solution exists for 80"m > 1.053, and tw o solutionsexist for 80~"3< 1.053.

- 6 - 4 - 2 0 2Fig. 9. Same as figure 8, for the case y = 0.5. No solutions exist for 50~" s > 1.077. Thepremixed flame character of the reaction zone for the Damkohler numbers close to theextinction one is already apparen t for this value of y.

(b) The re is leakage of both reactan ts through the flame as evidenced from thefact that (j8, - ) and (j8i + )_ are , functions of y and 8, different from zero. Fory > 1, (j8, + )_,*, is identically zero. For 7 = 0 , (jSi - !), and (jSi + ), are, of cours e,equal. When 7 increase from zero to on e, for fixed S0, (j3i + )-~ goes to zero and(0! - )=o goes to infinity.


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1028 A. LINAN

NEAR -EQUIL.DIFF. FLAM E STRUCTURE

(5^ e (/s-)ft

-^0-$y

0/

-4

Fig. 10. Diffusion flame structure for the case y 1. No D amkohler num ber of extinction exists in this case, as it occurs for \y\ > 1. There is not leakage of one reactantthrough one side of the flam e, (/3 + )_, = 0. For the lower v alues of 50, the reaction zonestructure is of the premixed flame type .

(fl-SJ.

Fig. 11. The nondimensional reactant leakage (j8 f ) , is plotted in terms of the reducedDamkohler number 80 for several values of y, in the near-equilibrium regime. The solution is found to be double-valued for 5 8B (y) if 7 < 1.

(c) The leakage (B t - ), is larger for the unstable branch than for the near-equilibrium stable bran ch. F or large 80 the solution corresponding to the unstablebranch shows a thin reaction zone, of the premixed flame type, at large negativevalues of .


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The asymptotic structure of counterflow diffusion flames 1029

(d) For y close to 1 the extinction regime occurs at small values of 0.5, and its error is less than one per cent for lower 7. For negative values of 7,calculate 80E using Eq. (82) with 7 replaced by \y\.The Damkohler number for extinction is obtained by replacing 8 by 80E in Eq.(74).However, the deviations of the temperature T from equilibrium, obtained in

Fig. 12. The first approximation for the Damkohler number of extinction 8B is shown asa function of |y|, as obtained from the numerical solutions (solid line) and as given by Eq.(82) (dashed line).


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1U30 A. LINAN

this near-equilibrium analys is, for small values of (1 - y) are of order Te2/Ta(l y). Hence, to predict the extinction condition for small values of (1 - y), theanalysis of the previous section should be used, unless the activation energybecomes sufficiently large.7. Discussion and generalization

The analysis of the previous sections covers all the regimes that can be foundwhen analysing diffusion flames, unless they are of the unsteady or evolutiontype [251, or the chemical reac tion is multi-step and do es no t admit a simplified d escription by a single overall irreversible reaction.The non-dimensional activation energy, or more precisely Ta/Tr2 (where T r isthe characteristic temperature of the reaction zone), has been used as the largeparameter in the asymptotic analysis. It is interesting to observe that the parameter TJTr is the product of the nondimensional activation energy andthe nondimensional heat release, or third Damkohler number, both based on thecharacteristic thermal energy at the temperature of the reaction zone.The first Dam kohler num ber, or nondimensional frequency factor, is allowedto grow approp riately w ith Ta/Tr2 so as to obtain n onzero heat release rate s in thelimit of infinite Ta/Tr2. In this limit the chem ical reaction take s place only at a welldefined temperature T r, which is then used in the analysis of the partial burningand premixed flame regimes as the independen t variable to calculate the corresponding Damkohler number D.As a result of the analysis the temperature and concentration distributionshave been obtained in term s of th e five param eters D, Ta, Tx, B and a for the fourpossible regimes. A detailed parametric discussion of the results will not beundertaken here. We shall give only a brief discussion of some of the results.In particular E qs . (41) and (59) prov ide th e relation s, for the partial burning regimes and for the prem ixed flame regim es, between the asym ptotic flame temperature and the Damkohler num ber.t Both curves, Tb(D ) and TP(D), have branchesthat exhibit an increasing temperature with decreasing Damkohler numbers. Astability analysis would very likely show these branches to be unstable.Both curves give an infinite Dam kohler num ber at a common transition temp erature. The analyses given for both regimes fail at temperatures close to thetransition temperature. An appropriate asymptotic analysis of a transition regimewould provide a smooth transition between the valid results of the partial burningand premixed flame regimes. Th e analysis is not included b ecause this transitionregime will alway s be un stable (as is also the case with the partial burning regime).Figure 13 shows the relation TP(D ) obtained from the asymptotic analysis ofthe premixed flame regime, for some values of the param eters and indicates howthe Damkohler number changes with changes in B and Ta. The lower branch ofthese curves is unstable. An extinction Dam kohler number can b e obtained fromcurves such as these as a function of B, a, T, and Ta.

fThis relation can be simplified by u sing the following approx imate relation for E = 2TT exp (z2) as afunction of x: E~ l = x2 In {(1 + cx)/2rrx2} for 0 < x =0.5 and the symmetric relation for 0.5 =s JC < 1,where c 3.938.


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r-__

(\

. . !

- r r - n n n i r i i r m rTo = 60T,>/ l)=.02

/(r /> \ C\ T a = 6 0 T ,* TaEBOT,,,

B = 0

i i l in n i i i m i l l

1 1 I 1 M il l 1 I 1 1M i l l

Ta=90Tco/ B = 0T Q =60T


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1032 A. LINAN

f t

tz

$ 8l

1**60%,

\Ta*l20TwV 0.01038a 0.061250'O

"JoinfiM.Present

Tg*120T~

9 K>" to" _ Kf tO U to $Fig. 14. The oxidizer consumption rate at the flame (- y'(0)la) is plotted in term s of theDamkohler number as obtained from the asymptotic analysis of the premixed flame regime, the partial burning regime, and the nearly frozen regime. The asymptoticDamkohler number of extinction is compared with the one obtained numerically by Jainand Mukunda.

112x100 = 1.79x10

T0 = 6 0 TooVl20Ta,TQ = infinite

0 = oo,Ta finiteV Ta = 0.06125j9=0

Fig. 15. Temperature distribution corresponding to two values of the heat flux from theoxygen-side, leading to the prem ixed flam e regime and th e partial burning regime.


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Williams [11] obtained a similar leakage of reactants through a near-equilibrium diffusion flame, associated with orders of the reaction with respect to the reactantsdifferent from unity.Figure 15 shows temperature profiles obtained by numerical integration fortwo oxygen-side fluxes, one corresponding to the premixed-flame regime and theother to the partial-burning regimes. Curves are given for three nondimensionalactivation energies, showing how the asymptotic structure described in Section 2is approached for very large activation energies. Again, rates of Jain andMukunda[6] have been used. The equilibrium solution also is included, so that thecorrespondence with Fig. 3 becomes evident. The curves demonstrate thickeningand some translation of the reaction zone.The character of the results demonstrates clearly that the analysis can beextended to other fluid dynamical configurations and to include more realistic descriptions of the transport coefficients. The structures of the outer zones willchange, but the key aspect of the analysis, namely the reaction-zone calculationsand matching techniques developed herein, will be identical.AcknowledgmentThis research was partially sponsored by the Air Force Office of Scientific Research through its European Office, under contract No. AFOSR 72-2254. Most of the numerical calculations that play an important role in this paper were carried out by Mr. V. Torroglosa. A number of theideas and concepts introduced in this paper became ordered and clear during many enjoyable hours ofdiscussion with Professor Forman Williams of the University of California S.D.; I am also in debt to himfor his encouragement in publishing this paper.Appendix A

We analyze here the ignition regime for the case where j8 is of order unity, orsimply 0 > TJlTa = e.It is clear from Eq. (6), written here in terms of the variablee = T-T + px, (Al)

asd26ldx2 = - 2TT exp (z2)y0yFD exp ( - TJTJ exp {Ta(0 - px)lTT}, (A2)

that the chemical reaction will be frozen except for small values of x, such as tomake px of order TJITa. Otherwise the Arrhenius exponential factor will becomeextremely small. Notice that in the ignition regime 0 is uniformly small, of order e.For the analysis of the reaction region we introduce the inner variableX = /3x/e (A3)

and expand T asT = Ta,+ e(el-x) + 0{elQne)}. (A4)We shall take into account that for small x, or x of order unity,2TTX2Z2 exp (z2) = (1 - 2/z2 +),

and2TT exp (z2) = j32(e*r2z-2 = j32(exrV2 , (A5)


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A. LINAN

(AT)

(A8)

where z2 is a function of e and f} satisfying the relationz2 = -ln(2ire2p~2z2), (A6)

so that for small e, z2 is a large number, z2->~ln(e2(3 ~227r).By substituting Eqs. (A3) to (A6) into (A2) we obtain the following equationfo r $i, if we neglect terms of order ze~2,

X2d2ejdX2 = - A(x ~ fi$i) exp (e t - x)where A is a reduced Damkohler number

A = -xzr 2aD exp ( - Ta/T)Equation (AT) is to be solved with the boundary conditions

0,(0) = 0, 0,*O) = O (A9)The last condition is obtained when matching the inner expansion (A4) with anouter expansion,

T=T~px +em-x) + -- , (AlO)for the frozen flow region. The matching conditions provide, in addition, therelation It - 0i(


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The asym ptotic structu re of counterflow diffusion flames 1035

two solu tions exis t for each A lower than an ignition Aj(j8) and no solution forA > Ax. We see reproduced in Fig. 16 the lower bend of the typical S curve s, relating maximum tem perature and D amkoh ler n um ber. Fo r /3 > 1, 0i(>) is uniquelyrelated to A.An asymptotic analysis of Eq. (A7) was carried out by Lilian and Crespo[25]for large values of 0i(), using the ideas of Section 5. For large 01() a thinreaction zone located around \r = /30i() separates a region of near-equilibrium,for lower , from a region of frozen flow for r > 0. This analysis can be usedto calculate the Dam kohler num ber of ignition for small values of (1 - )3), becausein first app roximation 0i() = 2/(1 - p) at the ignition conditions. The first twoterms of the expansion of A r in term s of (1 - /3) yield the following expression forthe Damkohler number of ignition

D I e - r / V = 2 e - 2 ( l - | 8 r 2 ( 2 ^ - j 3 2 ) (All)which surprisingly correlates very well the numerical results and has the correctasymptotic form for small values of p except for a factor 0.99. Fig. 16 includes thefirst term of the a sym ptotic relationsh ip be tween 0,() and A for large values of

The following expressionze2 = -In {87reV//3 2(l - pf) (A12)

should be used in Eq. (A ll) for ze instead of E q. (A6), so as to include the factor(1 - j3)~2 to accoun t for the fact that at small values of (1 - /3) the reaction zone iscentered around Xr = 2/(1 - p).How ever, it is clear that for sm all values of (1 - p) the ignition conditions result directly from the analysis of S ection 5, and E qs . (59) and (67) can be used topredict the Damkohler number of ignition with even better accuracy.

Appendix BIn the partial burning reg ime, the third term of the inner expansion 2 is givenby the equationd22ld2 = - A0exp2+ bx -WlTb -

d$ 2 /d = 0 for -*oo (B2)d2ld = B x for | - > - o o (B3)


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1036 A. LIN AN

A firs t integral of Eq. (Bl) , sat isfying Eq. (B2) is ,= - A 0 e x p 3>,{ - (3>,2 - 2


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The asym ptotic structure of counterflow diffusion flame s 1037

(C3) can only be calculated if we specify that the function y(rj) given by E q. (C3)be the first approximation y, in an expansion y = y, + my2 of the solution y ofEqs. (52)-(54) for small values of m, and that y2 be bounded. Then we obtainn = 1.344 and y,~ = 0 (C4)for small values of m.(b) Solution for m-*0.5When m approaches 0.5, both n and the leakage yi through the reaction zonebecome very large compared with one. The changes in y through the reactionzone remain of order unity so that in first approximation the concentration y ofthe reactant can be taken as constant in the reaction zone yi = y,, and then thereaction zo ne struc ture coincides with that of the partial burning regime. When ananalysis of Eq. (52) for small values of (0.5-m), or large values of y^, is carriedout following the procedure used in Section 4 we find in first approximation

y,co = ( l - 2 m r 1 , - n = ( l - 2 m ) - I + 2 1 n ( l - 2 m ) (C5)(c) Solution for large values of -mWe anticipate that in the reaction region y will be small, of order 1/m, so thatEq. (52) may be replaced in first approximation by the equation

2d2yldrj2 = y exp ( - mi)) (C6)which can be written as a Bessel equationy " + y ' / f - y = 0 ( C 7 )

in terms of the variable( = ( _ V 2 / m ) e x p ( - m T j / 2 ) (C 8).

Th e solution of Eq . (C7), finite for t -


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4 W J O A. LINAN

for large values of TJ (and t) so tha tyIOO = 0 (C14)

(d) CorrelationsThe asym ptotic solutions given abo ve have been used to generate the following correlations of the results of the numerical integrations of Eqs. (52)-(54).In the range - 0.2 < m < 0.5, a good approximate expression for nm is givenby rtm-1.344m-4m 2(l-m)/(l-2m) + 3 m 3 - l n ( l - 4 m 2 ) (C15)For small positive values of m and all negative values of m an approximateexpression for nm is

nm = - In (0 .6307m 2- 1.344m + 1) (C16)Relative errors in these formulas do not exceed a few percent.References

1. Spalding, D. B., Theory of Mixing and C hemical Reaction in the Opposed Je t Diffusion Flam e,ARSJ., 31, 763-771, (1961).2. Lifian, A. and Da-Riva, I., Non-equilibrium Effects in Hypersonic Aerodynamics, 3rd ICAS Congress, Stockholm (1962). The Proceedings of the Congress published by Spartan Books (1964).3. Fendell, F. E., Ignition and extinction on Combustion of Initially Unmixed Reactants, /. FluidMech., 21, 291-303 , (1965).4. Chung, P. M., Fendell, F. E. an d H olt, J. F. , Non-equilibrium Anomalies in the Development ofDiffusion Flames, AIAA J. , 4, 1021-1026 (1966).5. Fendell, F. E., Flame Structure in Initially Unmixed Reactants under One Step Kinetics, Chem.Eng. ScL, 22, 1829-1837, (1967).6. Jain, V. K. and Mukunda, H. S., On the Ignition and Extinction Problems in Forced ConvectionSystems, Int. Jour. Heat and Mass Transfer, 11 , 491-508 (1968).7. Jain, V. K. and M ukunda, H . S., Th e Extinction Problem in an Opposed Jet Diffusion Flame withCompetitive Reactions, Combustion Science and Technology, 1, 105-117, (1969).8. Sanchez Tarifa, C. Perez Del Notario, P. and Garcia Moreno, F., Combustion of Liquid Mono-propellants and Bipropellants in Dro plets, Eighth Sy mp. Int. Com bustion, Williams and Wilkins,Baltimore, 1035-1053 (1962).9. Peskin, R. L. and Wise, H., Ignition and Deflagration of Fuel Drops, AIAA J., 4, 1646-1650,(1966).10. Peskin, R. L., Polymerop ouious, C. E . and Y eh, P. S., Results from a Theoretical Study of FuelDrop Ignition and Extinction, AIAA J., 5, 2173-2178, (1967).11. Kassoy, D. R., and W illiams, F. A., Effects of Chemical Kinetics on Ne ar Equilibrium C omb ustionin Nonpremixed Systems, Phys. of Fluids, 11 , 1343-1351, (1968).12. Kassoy, D. R., Liu, M. K. and Williams, F. A., Comments on Effects of Chemical Kinetics onNear-equilibrium Combustion in Non-premixed Systems, Phys. of Fluids, 12, 265-267, (1969).13. Zeldovich, Y. B., On the Theory of C ombustion of Initially Unmixed Ga ses, NACA Tech . Mem.,No 12% , (1951).14. Friedlander, S. K. and Ke ller, K. H., The Struc ture of the Zon e of Diffusion-Controlled Chem icalReaction, Chem. Eng. ScL, 18, 36 5, (1963).15. Kirkby, L. L. and Schmitz, R. A., An Analytical Study of the Stability of a Laminar DiffusionFlame, Combustion and Flame, 10, 205-220, (1966).16. Schmitz, R. A., A Further Study of Diffusion Flame Stability, Combustion and Flame, 11,49-62,(1967).


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17. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Chapter VI., PergamonPress, (1969).18. Burke, S. P. and Schu mann, T. E. W ., Diffusion Flam es, Ind. and Eng. Chemistry, 20, 998-1004,(1928).19. Williams, F. A., Combustion Theory, Addison Wesley, (1965).20 . Librovitch, V. B., Personal communication, (1970).21. Linan, A. On the Internal Structure of Lam inar Diffusion Flame s, OSR/EOAR, TN 62-69. INTA ,M adrid, (1961).22 . Linan, A., On the Structure of Laminar Diffusion Flames, Tech. Rep t. FM 63-2 . INTA , Madrid,(1963).23 . Linan, A., Diffusion Flames and Supersonic Co mbustion, An nu. Sci. Rept. No 1. EOAR C ont. NoF61052-69-C-0036. INTA, Madrid (1970).24. Libby, P . A., and Eco nom os, E., A Flame Zone M odel for Chem ical Reaction in a Laminar Bound

ary Layer with Applications to the Injection of Hydrogen-Oxygen Mixtures, Int. J. Heat MassTransfer, 6, 113-128 (1963).25. Linan, A., and Crespo, A., An Asymptotic Analysis of Unsteady Diffusion Flames for LargeActivation Energies, Tech. Rep. for ARO-INTA Subcontract, Madrid (1972).

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