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Unformatted text preview: FLAME IGNITION Paul D. Ronney Department of Aerospace and Mechanical Engineering University of Southern California, Los Angeles, CA 90089 Abstract Current understanding of the initiation of deflagration waves from electric sparks and other concentrated sources of energy is reviewed, emphasizing understanding obtained since the seminal works by Lewis and von Elbe. The primary focus is on gasphase phenomena but ignition of aerosols and dust suspensions is also discussed. Initially, a phenomenological approach to used to describe basic ignition phenomena. Then, starting from a simplified set of conservation equations for a onedimensional system with spherical symmetry, the phenomenological approach is refined using recent theory based on activation energy asymptotics. These models, and computational extensions, are compared with experimental observations. Influences of the state of the combustible mixture, the characteristics of the ignition source, and the flow environment on these ignition processes are examined. It is shown that simple diffusivethermal models are adequate to qualitatively describe many of these influences, especially when Lewis number effects are included. Applications to combustion systems of practical importance, e.g. internal combustion engines and fire safety considerations, are discussed and topics requiring further study are identified. AME514 FALL 2004 INCOMPLETE DRAFT VERSION NOT FOR ATTRIBUTION 1 TABLE OF CONTENTS NOMENCLATURE............................................................................................................3 1. INTRODUCTION..........................................................................................................4 2. BASIC CONCEPTS OF FLAME IGNITION.....................................................................2 2.1 Phenomenological description of Lewis and von Elbe..........................................2 2.2 Estimates of the minimum ignition energy..........................................................2 3.0 MATHEMATICAL THEORY OF FLAME IGNITION......................................................5 3.1 The conservation equations for flame initiation.................................................5 3.2 Solutions using activation energy asymptotics....................................................7 3.2.1 Steady solutions.................................................................................7 3.2.2 Stability of the stationary solutions and application to ignition .......10 3.2.3 Timedependent solutions...................................................................11 3.3 Numerical Solutions..........................................................................................12 3.4 Detailed versus simplified chemical kinetics and transport ...............................12 3.5 Multidimensional effects..................................................................................13 4.0 DYNAMICS OF IGNITION PROCESSES......................................................................15 4.1 Subcritical ignition kernels..............................................................................15 4.2 Transition from ignition to steady propagation...................................................16 5.0 EFFECTS OF THE STATE OF THE COMBUSTIBLE MIXTURE .......................................17 5.1 Pressure ............................................................................................................17 5.3 Stoichiometry...................................................................................................17 5.4 Additives.........................................................................................................18 6.0 EFFECTS OF THE CHARACTERISTICS OF THE ENERGY DEPOSITION SOURCE.......18 6.1 Volume.............................................................................................................18 6.2 Duration...........................................................................................................19 6.3 Method of energy deposition..............................................................................20 6.3.1 Spark discharges...............................................................................20 6.3.2 Thermal energy versus radicals..........................................................20 6.3.3 Lasers ................................................................................................20 7.0 EFFECT OF FLOW ENVIRONMENT............................................................................21 7.1 Mean flow and mean strain ...............................................................................21 7.2 Turbulence........................................................................................................21 9.0 IGNITION OF HETEROGENEOUS MIXTURES............................................................21 9.1 Aerosols...........................................................................................................21 9.3 Nonpremixed systems......................................................................................22 10.0 PRACTICAL APPLICATIONS....................................................................................22 11.0 CONCLUSIONS.........................................................................................................23 REFERENCES....................................................................................................................23 2 NOMENCLATURE A Cp D dq Ea Eign Emin H H chem k Le P q Q r R Rf Rz SL T t U v y Y a b c g d e f h q r s t W Preexponential factor in reaction rate expression Specific heat at constant pressure Mass diffusivity Quenching distance Activation energy Ignition energy Minimum ignition energy Heat release per unit mass of fuel Chemical enthalpy release before flame extinguishment Thermal conductivity Lewis number Pressure Scaled ignition power (Eq. 15) Energy input Spatial coordinate Nondimensional spatial coordinate Flame front radius Zeldovich (stationary) flame radius Ideal gas constant Laminar burning velocity Temperature Time Nondimensional velocity (Eq. 7) Fluid velocity Mass fraction of scarce reactant Mass fraction of scarce reactant scaled by ambient value (Eq. 7) Thermal diffusivity Nondimensional activation energy (Eq. 7) Scaled flame radius (Eq. 14) Gas specific heat ratio Flame thickness Nondimensional temperature ratio (Eq. 7) Fuelair equivalence ratio Geometrical parameter (Eq. 6) Nondimensional temperature (Eq. 7). Density Scaled time (Eq. 16) Nondimensional time (Eq. 7) Nondimenionsal energy input (Eq. 7) Subscripts and superscripts ad o * (` ) Value at adiabatic flame conditions Value at ambient conditions Value at surface of Zeldovich flame Temperatureaveraged values 3 1. INTRODUCTION Ignition is defined as the transformation of a combustible material from an unreactive state to a selfpropagating state where the ignition source can be removed without extinguishment resulting. The understanding of flame ignition processes is of critical importance in most energy conversion systems and in particular reciprocating internal combustion engines because combustion events must be initiated a t rates up to 200/sec in each combustion chamber. This understanding is also of critical importance in fire safety considerations to delineate hazardous from harmless environments. Flame ignition processes should not be confused with chemical ignition [9930] which indicates the transformation of an homogeneous mixture from a slowly reacting state to an accelerating reaction as a result of chain branching or thermal feedback. As the name would suggest, in chemical ignition, the interaction of different chemical reactions and radical loss mechanisms plays the dominant role and transport processes play a minor role. While understanding of chemical ignition processes are of considerable practical importance, for example in engine knock, it is of less value for deliberate initiation of reaction in practical combustion devices because the entire volume of combustible material, rather than a small region, must be influenced by the ignition source. Invariably ignition systems for combustion engines take advantage of the propagation of deflagration or detonation fronts, which are fundamentally dependent on transport effects, to minimize the energy required for ignition. Thus, in this review, homogeneous ignition will not be discussed further. Ignition limits should also be distinguished from flammability limits [9931], which delineate mixtures which cannot exhibit indefinite selfpropagation, regardless of the ignition source, from those which can. Given these definitions, clearly ignition limits will in general exist for mixtures which are flammable, whereas ignition limits do not apply to mixtures which cannot exhibit indefinite selfpropagation. This review will focus on the ignition of premixed combustible gases, but ignition of heterogeneous mixtures (sprays, dust suspensions, etc.) will also be discussed. Both the critical conditions for ignition and the dynamics of the ignition process will be addressed. Attention will be focussed on the initiation of deflagrations (subsonic combustion waves) rather than detonations (supersonic waves), as the two are very different processes; the latter is primarily a function of t h e production and interactions of gasdynamic waves and locally homogenous chemical reactions. Reviews of the initiation and propagation of detonation waves are given by Lee [9932, 9933]. The format of this review is as follows. In section 2, the phenomenological concepts of ignition are introduced. In section 3 and 4 these concepts are used to motivate and develop a mathematical framework from which to describe ignition processes. Both analytical and computational approaches are discussed and the limitations of each are identified. In sections 5 through 9 the effects of physical 4 parameters on ignition processes are examined. In section 10, the applications of this understanding to practical combustion devices is considered. Concluding remarks are given in section 11 and topics warranting further study are identified in section 12. 2. BASIC CONCEPTS OF FLAME IGNITION In this section the basic processes which occur in flame ignition are described. It is shown t h a t very elementary models can provide a reasonably good description of many aspects of flame ignition. The limitations of these models are discussed and from these limitations the motivation for t h e development of more rigorous models of ignition are given. 2.1 Phenomenological description of Lewis and von Elbe Lewis and von Elbe [991] describe ignition of premixed gases as follows (Fig. 1): if a subcritical quantity of energy in the form of heat and/or radicals is deposited in a combustible mixture, t h e resulting flame kernel decays rapidly because heat and radicals are conducted away from the surface of the kernel and disociated species recombine faster than they are generated by chemical reaction in t h e volume of the kernel. The kernel extinguishes after consuming a small quantity of reactant. This extinguishing kernel is sometimes called a nonignition. On the other hand, if the ignition energy exceeds a certain threshold {called the minimum ignition energy (Emin)}, at the time when the peak temperature decays to the adiabatic flame temperature (Tad) the temperature gradient in the kernel is sufficiently shallow that heat can be generated in the kernel faster than it is lost due to conduction to the unburned mixture. This leads to the development of a steady flame which consumes all of t h e available mixture. This description indicates that flame ignition is generally characterized by a minimum total energy rather than a minimum energy density. Hence, a primary goal of flame ignition studies, and a focus of this review, is to determine Emin as a function of the properties of the combustible mixture and the characteristics of the ignition source. The phenomenological description dictates that this energy be deposited over a sufficiently short duration and small volume, as discussed in section 6.1, otherwise the energy will be too sparsely distributed to raise the gas temperature sufficiently to cause rapid chemical reaction and heat release. 2.2 Estimates of the minimum ignition energy According to the above hypothesis, Emin can be estimated in the following way. The minimal radius of the developing flame kernel is expected to be related to the laminar flame thickness (d). This 2 is because d is determined by the balance between heat generation via chemical reaction and h e a t conduction to the unburned gas, which is functionally the same balance as just discussed in relation to ignition kernels. d can be estimated making the plausible assumption [993] that the enthalpy convected through the flame front is comparable to the enthalpy conducted from burned gas to unburned mixture, which can be written as r o SLCp (Ta d  To ) A f A f Ta d  To d (1) where Af is the crosssectional area of the flame. Hence d= k k ao k = ;a o o ro C p SL ko SL r oC p (2) Since, approximately, k ~ T0.75, for flames with a typical Tad/To = 7, d4 ao SL (3) Then the enthalpy contained in the prescribed volume is Emin k (T  T ) 4p 3 4p d rCp (Ta d  To ) 0.3 d 3r o Cp (Ta d  To ) 34a 2 o a d3 o o 3 3 SL (4) The most important point to be gleaned from Eq. (4) is that Emin does not scale simply with t h e fuel concentration, equivalence ratio, or SL . Among the interesting predictions of these equations are: 1) More reactive mixtures (higher S L ) will have much lower Emin. This is consistent with experimental findings. For most fuels in air, a plot of S L vs. f has a maximum near stoichiometric conditions (f = 1), and thus a plot of Emin vs. f should be and is Ushaped with a minimum near f = 1. However, Lewis number effects (section 5.2) shift the location of the minimum relative to f = 1. 2) Since a ~ P1, where P is the pressure, for cases where SL is independent of pressure, Emin ~ P2. This is close to what is found experimentally [991] for nearstoichiometric hydrocarbonair mixtures where the overall reaction order is close to 2 [9937] and thus S L is independent of pressure. 3 3) Since a and k are determined primarily by the diluent gas, e.g. nitrogen in fuelair mixtures, Emin would be about 100,000 times larger in a Hediluted mixture than an S F6 diluted mixture having the same burning velocity at the same pressure. {It should be realized, however, t h a t He has a much lower Cp than S F6, thus much higher values of Tb at the same diluent concentration and therefore SL than SF6. Hence, the SF6diluted mixture would require a much lower diluent concentration to obtain the same SL.} This is contrary to one's desire to associate higher transport rates in He with greater ignitability. Evidence for the prediction is seen in a comparison of a (fuellean) 10%H2 18.9%O2 71.1%He mixture and a 10%H2 18.9%O2 71.1%Ar mixture, in which the Ardiluted mixture has an Emin 12 times lower than the Hediluted mixture [991], despite the fact that SL is higher with He dilution. 4) Since Emin ~ rCp(TadTo)d 3 , r ~ P, and the quenching distance (dq) of a combustible mixture is proportional to d [993], it would be expected that
3 Emin Kdq P; K = constant (5) for mixtures with fixed molecular weight and Cp (i.e. for a given diluent). In this context, Lewis and von Elbe [1] show a plot of Emin versus dq for many fuelair mixtures at varying pressures (their Fig. 176) which shows a slope of 3 at small dq, but the plot bends toward smaller slopes at larger dq, corresponding to lower pressures. They assert that the estimate Emin ~ dq3 fails a t large dq because heat losses to spark electrodes is more significant at large dq. However, they did not account for the pressure effect that appears in Eq. 5 even though their data were obtained at pressures between 0.1 and 2.5 atm. When their data are replotted as a function of dq3 P (Fig. 2), the results are more satisfactory in that a single line of constant slope fit all t h e data well, though ideally the slope of the bestfit line in Fig. 2 should be unity. It should also be noted that of the most notably outlying data points, generally points below the bestfit line are lean H2 or lean CH4 points, and point above the line are lean C3 H 8 points. This is likely due to Lewis number effects, discussed in section 3.2.1. 5) For the representative case of a stoichiometric CH4 air mixture at 1 atm (ao = 0.2 cm2 /sec, k o = 0.026 W/mK, Tad = 2200K, SL = 40 cm/sec) Eq. (4) predicts Emin 0.010 mJ. This is about 30 times lower than the experimental result [991] (see Table 1).. This discrepancy is a function of several factors including chemical kinetics (section 3.4), heat losses (sections 6 and 8) and possibly shock losses (section 6.1). 4 6) On the other hand, for a 10% H 2 air mixture ( SL 10 cm/sec [995]), Eq. (4) predicts Emin 0.3 mJ, which is 2.5 times higher than the experimental value [991]. Furthermore, for two mixtures with nearly the same a o, ko, Tad and S L , Emin should be similar, for example hydrocarbonair mixtures with the same S L ; however, experiments show that a lean CH4 air mixture with S L 5 cm/sec has an Emin about 1000 times lower than a lean C3 H 8 air mixture with the same S L and thus d (5 mj [992] vs. 5000 mj [999]). According to Eq. (4), the lean CH4 air and C3 H 8 air mixtures should have the same Emin. We shall show that much of these discrepancies is a result of Lewis number effects. While the phenomenological model is able to describe qualitatively many of the important features of ignition, to obtain a more quantitative description the phenomenological model of ignition must be abandoned in favor of a more rigorous description of the heat and mass transport occurring in ignition processes. 3.0 MATHEMATICAL THEORY OF FLAME IGNITION We start with the simplest formal approach which seems capable of providing a satisfactory description of ignition processes. Results of such a model are given, and the effect of refinements of t h e chemical and transport models are indicated. 3.1 The conservation equations for flame initiation Given the following assumptions: The system is onedimensional The system can be treated as a mixture of ideal gases The system is adiabatic except for the ignition source All but one of the reactants are present in large excess, so that only a single, scarce reactant need be considered Chemical reaction can be modelled by a single irreversible step that is firstorder in t h e scarce reactant The values of the transport coefficients rD and k are constant The values of all thermodynamic properties (except temperature, composition and density) are constant The Mach number is sufficiently small that the momentum equation reduces to a statement of constant pressure 5 No body forces the basic equations for conservation of mass, energy, species and equation of state describing t h e evolution of a flame in a field containing a source of energy for ignition Q(r,t) (but no other sources or sinks) may be written as r 1 h + (r rv ) = 0 t r h r T 1 h rCp + rCp h (r vT ) = rkh r r h T ^ + rHyA exp r t r r y 1 rD h y ^ E r + rv h ( rh y ) = h r  ryA exp Ta t r r r r r (6a) ( ) + Q(r, t)
 Ea T (6b) (6c) (6d) ( ) rT = r o To = constant Here h is a geometric parameter; 0, 1, and 2 correspond to planar, cylindrical, and spherical geometry, respectively. defined: Following Frendi and Sibulkin [994], the following nondimensional variables are q T Ae b ; t tAe b ;R r ;U Ta d ao v a o Ae b ;b Ea Ta d
(7), e To y k Q(r, t) ;Y ; Le ;W Ta d yo rCp D r o Cp To Ae  b where Tad = To + yoH/Cp is the adiabatic flame temperature, these equations may be written as (1/ q ) 1 h 1 ^ + h R U =0 t R R q q 1 1 +U h ( Rh q ) = q R h R R h q ^ + (1 e )Yexp t R R e R Y 1 1 1 +U h ( R hY ) = Le q Rh R Rh Y ^  Yexp R t R R e (8a) ( b ( )) + W(R, t )q
1 1 q (8b) ( b ( ))
1 1 q (8c) The initial and boundary conditions are those of an initially quiescent, cold, unreacted, infinite medium, i.e. 6 q (R, 0) = e ;Y (R, 0) = 1;U(R, 0) = 0 for all R q (R, t ) = e ;Y(R, t ) = 1;U(R, t ) = 0 as R q Y U = = = 0 at R = 0 and as R R R R
(9). It can be seen that this highly schematic model of flame initiation consists of three equations which must be solved for q, Y, and U as functions of R and t for various values of the three parameters Le, b, and e and the characteristics of the ignition source (R,t). Even this simplified set of equations is in general beyond the reach of analytical techniques and can be difficult to solve numerically. In t h e following sections some of the solutions to Eqs. (8) and (9) are discussed. 3.2 Solutions using activation energy asymptotics 3.2.1 Steady solutions Zeldovich [996] showed that equations (8) and (9) in spherical geometry (i.e. h = 2) admit an adiabatic steady solution with a stationary flame radius R z . Such structures have been termed "Zeldovich flames." Rz is an eigenvalue of the steady system and can be thought of as an analog of burning velocity for planar propagating flames. The relevance of this solution to ignition will become evident later; intuitively one would expect Emin ~ Rz 3 . While it may seem at first surprising t h a t stationary, nonpropagating spherical premixed flames can exist, the solutions are easily obtained by setting (R,t) and the time derivatives to zero. Solving (8a) we find that the only acceptable solution is U 0, i.e. no convection anywhere in the flow field. For reactants with high activation energy, i.e. b , the technique of activation energy asymptotics (AEA) is frequently employed to obtain analytical solutions. For large b, the chemical reaction terms in (8b) and (8c) are negliglible except in a thin (relative to Rz ) zone near the peak temperature in the domain. Then for the chemically inert zones (8b) and (8c) become h q ^ h Y ^ R = 0; R =0 R R R R (10) which have solutions that are bounded as R only for spherical geometry (h = 2), in which case, upon applying the boundary conditions, the solutions are 7 CY > R + e; Y = 1 R R < Rz: q = Cq < ; Y = CY < ; Cq> ,CY > ,Cq < ,CY < constants R > Rz: q = Cq > (11). No nontrivial steady solution exists in cylindrical (h = 1) or planar (h = 0) geometry, there being no way to satisfy the boundary conditions as R . This is essentially the same situation as that of steady heat transfer from a source in an infinite medium; in spherical geometry the steady Nusselt number is 2, whereas in planar or cylindrical geometry there is no steady solution; the Nusselt number is inherently unsteady. To evaluate the constants Cq and CY, note that Eqs. 8b and 8c can be combined to eliminate the reaction terms and solved to obtain q+ (1  e )Y 1e =e + = constant Le Le (12). If Rz is defined as the radius where the temperature reaches its maximum value, i.e. at the reaction zone, then (11) becomes 1  e Rz + e ; Y = 1  Rz R Le R R < Rz : q = q * = constant; Y = 0 R > Rz : q =
where (13). q* T* 1 e =e+ Ta d Le (14). It should be noted that the temperature outside the reaction zone decays in proportion to R z /r, whereas for a propagating naerlyplanar flame temperature decays as exp(x/d). The difference between 1/r and exponential decay is very significant, because the sensible energy at r > R z , which is proportional i4r2 (1/r)dr , is infinite, whereas for the plane flame, iexp(x/d)dx is finite. Thus for t h e to
Rz
0 spherical steady flame, the behavior of the chemicallyinert farfield are much more important to determining the dynamics of flame evolution than in the case of nearlyplanar flames. The most important consequence of this solution is that in the fully reacted gas (Y = 0), t h e dimensional temperature (T* ) is given by 8 T * = To + Ta d  To Le (15). Note that T* > Tad for Le < 1 and T* < Tad for Le < 1. Because b is large for most practical combustion systems, small departures of Le from unity causes a large change in reaction rate. This will in turn cause a large increase (decrease) in the heat release rate at the reaction zone of the stationary flame for Le less (greater) than unity. This leads to a decrease (increase) in Rz because it affects the temperature gradient which the reaction zone can support at steadystate. These theoretical results are interpreted in the following physical sense. The Lewis number (Le) is defined here as Thermal diffusivity of the bulk mixture Mass diffusivity of the scarce reactant into the bulk mixture Le (16) This parameter can be interpreted as the ratio of the rate of diffusion of thermal enthalpy of the bulk mixture to that of chemical enthalpy of the unburned reactants. When Le differs from unity, flame front curvature, strain, unsteadiness, etc. cause changes in the rates of transport of chemical and thermal enthalpy which in turn affects the temperature at the flame front [993]. For flame in mixtures with low Le that are concave towards the burned products (as in an expanding spherical flame) t h e increase in the rate of chemical enthalpy to the flame front is greater than the increase in the rate of thermal energy loss, and thus the curved flame will burn more intensely than a planar flame in t h e same mixture. Since heat release reactions in most combustible mixtures have high activation energies, these changes in flame front temperature lead to large changes in reaction rate at the flame front and thus the local, instantaneous propagation rate. This could potentially explain the difference between Emin for the lean methaneair and propaneair mixtures mentioned above  according to Eq. 16, Le 0.95 for lean methaneair and Le 1.7 for lean propaneair mixtures. The above definition of Le presumes that one of the reactants is greatly deficient (e.g. a very lean or very rich fuel/air mixture.) For mixtures near stoichiometric, diffusion of two or more reactants must be considered, as discussed in section 5.2. Using activation energy asymptotics, for general Lewis numbers Rz is found to be [997, 998] Rz = d b 1 exp *  1^ ^ 2 q Le (17) where d k/roCpS L as before and SL determined from a separate AEA analysis [993, 9938] which yields to leading order in b1 9 SL = 2 eLea o A b ^ exp 2 b (18). Equation (17) is a key result, for it shows the importance of Le. Note that Rz = d when Le = 1, and with the typical values b = 10 and e = 1/7, for Le = 0.5, R z = 0.11d and for Le = 2, R z = 37d. Thus, Le has a profound effect on Rz under realistic conditions. 3.2.2 Stability of the stationary solutions and application to ignition Linear stability analysis of the steady solutions [997, 998] confirms the expectation that stationary spherical flames are unstable, i.e. if the flame at its equilbrium radius R z is perturbed to an infinitesimally smaller value it will continue to collapse (presumably to extinguishment, although this cannot be determined by linear stability analysis) whereas if it is perturbed to a radius larger than R z it will continue to expand (and presumably grow to a normal flame.) This suggests that the flame initiation criterion will be linked to the production of a hot ball of gas of radius R z rather than d. Hence, Rz can be thought of as a critical flame kernel radius for ignition. {These stability predictions are altered markedly when heat losses from the ignition kernel are considered, as will be discussed in section 8.} These results indicate that the previous estimate of Emin, Eq. (4), is probably valid only for Le 1 (Rz d), if at all. Since, roughly, we expect Emin ~ Rz 3 , it is clear that the ability of point sources of energy to ignite a combustible mixture is profoundly affected by its Lewis number. This is true even for mixtures with the same thermal diffusivity and planar burning velocity, i.e. the same d. This point is illustrated in Fig. 3, which shows (Rz /d)3 as a function of Le for fixed values of e and b. As a specific example, consider the previously cited lean CH4 air and C3 H 8 air mixtures which both have S L 5 cm/sec and Tad 1550K, yet their values of Emin differ by a factor of 1000. For the C3 H 8 air mixture Le 1.78 and Ea 38.7 kcal/mole, whereas for the CH4 air mixture Le 0.96 and Ea 43.6 kcal/mole [9951]. According to Eq. 17, the ratio of R z 3 for the two mixtures is about 3700, which is of the same order of magnitude as ratio of Emin. It is also worth noting that is was not necessary to invoke the assumption of constant density, frequently employed in analyses of this type, to obtain either the steady solutions or the stability results. In the former case this is because the continuity equation reduces to the statement U = 0, and in the latter case because, to leading order in the asymptotic expansion, convective transport is negligible compared to diffusive transport when Le < 1. 10 3.2.3 Timedependent solutions While the above analysis indicates the important role of the steady spherical flame in ignition processes, it does not show that deposition of energy equivalent to that inside a ball of radius R z is an ignition criterion [998]. A fully timedependent model which considers the profile of energy input (as opposed to the linear stability analysis discussed above) is needed. In general this problem is mathematically intractable, but Joulin [9910] has shown that it can be solved for the case of Le less than and bounded (in the asymptotic sense) away from unity, i.e. Le  1 = O(1), not O(b1) . Additionally, constant density must be assumed and thus the idealgas equation of state is ignored. For this case a parameterfree evolution equation for the flame front radius has been obtained [9910]: c (s )ln (c (s )) + q(s ) dc(s) = c ( s ) 2 ds 0 s ds ss (19) where c Rf/Rz , where Rf is the instantaneous flame radius, q is the nondimensional ignition power given by q Q 4 pkRzTa d (q * )
2 (20), Q is the dimensional ignition power applied at R = 0 (not to be confused with the Q, the ignition power per unit volume) and s is a scaled dimensionless time given by (q * )2 Le ^ at s 4 p ~ 2 1 e 1  Le Rz 2 (21). Examples of numerical solutions of (19) for different q(s) are shown in Fig. 4. It is seen that c = 1 (i.e. r = R z ) is a saddlelike point from which Rf eventually departs. For values of ignition energy greater than the critical value Ec, the flame expands indefinitely outward and for values of ignition energy less than Ec, the flame quenches. These calculations lead to the following relation for Emin at the optimal duration for ignition (discussed more fully in section 6.2): 1 e ^ 1  Le ^ ~ ( ) 3 Emin 14b q * Le r a dCp Ta d  To Rz e 2 (22) 11 Equation (22) has the expected form, in particular Emin ~ radCp(Tad  To)Rz 3 . The factor of b is somewhat surprising; since the analytical results were obtained in the limit b , it indicates t h a t while Rz is the appropriate length scale, an asymptotically large multiplicative factor is also present. Practically speaking, though, b does not vary substantially for common combustible mixtures (typically b 10  15) and so plays no substantial role in the estimate of Emin in Eq. (22) {but note the powerful effect of b on Rz , Eq. (13).} For the 10% H2 air mixture (Le 0.35) mentioned in section 2.2, using Ea 27,000 cal/mole [9939] along with temperatureaveraged transport properties, Eq. (22) yields Emin 0.011 mj. This is a factor of 15 lower than the experimentally determined value of 0.16 mJ [991], hence, consideration of Lewis number effects reduces Emin far below the previous estimate (Eq. 4), and is now below t h e experimental result as expected. The timedependent analyses for Le 1 or variable density analogous to the constantdensity, Le < 1 case just discussed cannot be reduced to a relatively simple evolution equation such as Eq. (19), and thus the analysis must be performed numerically starting from Eqs. (8). described in the following section. 3.3 Numerical Solutions In references [9911] and [9912], the analysis of the previous section are extended to Le 1 and variable density, respectively. Rz was again used as the reference length scale, and a mathematically pointsource ignition scheme was employed The results are quantitatively different, though not very different, from the previous analysis. Most importantly, the concepts of the critical flame radius and the role of Le discussed above still apply, though of course variable density effects must be considered to obtain quantitatively accurate results. Several other studies [9950, 994, 9925, 9952, 9925] have employed primitive variable formulations without using Rz as a reference length. the concept of a critical flame radius. Evolving temperature profiles qualitatively similar to that shown in Fig. 1 are found. All report importance of These factors indicate the validity of the semianalytic approach of Joulin and collaborations [9910, 9911, 9912]. 3.4 Detailed versus simplified chemical kinetics and transport Table I shows a comparison of values of Emin for stoichiometric CH4 air and C3 H 8 air mixtures from these computational studies, along with the predictions of a detailed numerical study and corresponding experimental results. It can be seen that, with one exception [9950], the simple phenomenological model, Eq. 4, and the numerical studies employing simplified chemical and transport models yield similar predictions of Emin. However, these values are systematically much lower than These analyses are 12 either experiments or the predictions of the study employing detailed chemical and transport models. Consequently, while the simplified chemical models are adequate to describe many of the important qualitative features of ignition processes, they do not appear to be adequate for quantitiative accuracy, even though the onestep models employed [9948, 9953] are able to predict correctly the planar steady burning velocities of CH4 air and C3 H 8 air mixtures. Motivated by the discrepancies between experiments and the simplied models, Sloane and Ronney [9940] compared Emin for a stoichiometric CH4 air mixture using two onestep chemical mechanisms [9948, 9953] and a detailed mechanism [9955], as well as simplified and detailed thermodynamic and transport models. It was found that only the differences in chemical mechanisms had a substantial effect on Emin. The most important reason for this is that most simplified chemical mechanisms are designed to reproduce the steady burning velocities of premixed flames, or extinction strain rates of nonpremixed flames. In either case, the conditions at the reaction zone of the flame is characterized by high temperatures, low reactant concentrations and significant concentrations of radicals. radicals. This is very different from the conditions in a developing ignition kernel  a low but As a consequence, it was found that the simplified chemical mechanisms greatly increasing temperature, reactant concentrations near the initial values and low concentrations of underestimated the homogeneous ignition delay time at the temperatures relvant to a developing ignition kernel. It was concluded that, at a minimum, a simplified chemical mechanism appropriate for studying flame ignition must be able to reproduce the homogeneous ignition delay time a t temperatures comparable to the adibatic flame temperature, as well as reproducing the steady planar burning velocity. Using a detailed chemical mechanism, and taking account of the size of the energy deposition region on Emin (section 6.1), Sloane and Ronney predicted that for a stoichiometric CH4 air mixture at 1 atm, Emin should be about 0.10 mJ. The lowest experimental value reported by Lewis and Von Elbe is 0.33 mJ. These two figures are fairly close when the likely effects of heat losses to the electrodes and gasdynamic shock losses in the experiments are considered (section 6.1). Thus it appears that, at least for fuels having chemical mechanisms that are reasonably well known, detailed modelling may provide a reasonable estimate of Emin (with some caveats concerning the interpretation of experiments, as discussed in sections 6.1 and 12). 3.5 Multidimensional effects Up to this point, only models employing spherical symmetry have been considered. In practice, electric sparks and other common ignition sources are hardly spherical. Thus, it is important to ascertain whether ignition phenomena modeled in the context of spherical sources are relevant to practical ignition systems. A priori, one might expect that if the dimension of an asymmetric source 13 were small compared to Rz , the ignition characteristics would be unchanged because these asymmetries would be softened by diffusion of the thermal energy, the rate of which is expected to be inversely proportional to the square of the feature size. Essentially this view was advanced by Joulin and collaborators [9910, 9911, 9912] in their theory of pointsource ignition. For ignition by electric sparks, a quasicylindrical spark is created across the spark electrodes. Several studies [9916, 9911, 9956] have shown that the rapid deposition of energy typically causes a cylindrical shock wave to form, which in turn generates a vortexring flow pattern that persists long after the shock wave has propagated outside the region which can influence the ignition process (Fig. 5). It is difficult to determine from the photographs of this process whether this flow pattern has a substantial effect on Emin or other ignition parameters. Perhaps with this motivation, Kono and collaborators [9929] numerically studied the ignition of C3 H 8 air mixtures at f = 0.75 using a twodimensional axisymmetric model that included gasdynamic effects including shocks. Their model presumed symmetry about the plane bisecting t h e spark gap, and so could not reproduce asymmetries about this plane which are found to occur for DC sparks [9916], presumably due to the impulse of fluid momentum arising from the net migration of electrons from the negative to positive electrode. The computed flow patterns are qualitatively Quantitatively, the values of Emin similar to those found in experiments employing AC sparks. computed are more than a factor of 100 lower than the experimental value (0.015 mJ vs. 2.5 mJ at t h e optimal spark duration), which would seem surprising considering that these calculations included t h e effects of heat losses to the electrodes as well as shock losses (see section 6.1). It is difficult to know how much of this discrepancy is due to multidimensional effects because the did not compare their results to a onedimensional calculation. However, it it likely that much of the discrepancy is a result of the transport model assumed. In particular, these authors assumed unit Lewis number, whereas for the lean propaneair mixtures studied, Le 1.7. As discussed in section 3.2.1, this difference can have a very large effect on Emin. In fact, their predicted values of Emin are much closer to those found experimentally for lean methaneair mixtures at f = 0.75 (0.45 mJ [1]), which have values of Le much closer to unity. At low lewis number, it is well known that diffusivethermal instabilities lead to t h e formation of multidimensional cellular flame structures [993, 9957]. A priori, it is possible that this could affect Emin. However, it has been shown [9962] that the unstable Zeldovich flame shown here to play a key role in describing ignition is stable to threedimensional disturbances. Thus, while no multidimensional studies of ignition at low Le have been reported, at least for the purposes of computing Emin, these diffusivethermal effects are probably not significant. However, the cellular structure is very important to determining the extinction limits of these mixtures, because the breakup of the flame front into cells enables the temperature of the flame at the nose of these cells to be greater than Tad 14 [993, 9957, 9958], which in turn enables cellular flames to exist in mixtures that could not support a plane flame due to heat losses [9959, 9960]. 4.0 DYNAMICS OF IGNITION PROCESSES The main factor considered thus far in the discussion of flame initiation is Emin. However, other characteristics of ignition processes are also of interest. In this section, the dynamics of t h e ignition kernel are discussed, including the eventual fate of flame kernels resulting from subcritical ignition sources and, for successful ignition, the transition to steady flame propagation. 4.1 Subcritical ignition kernels As discussed in section 1, subcritical ignition sources produce flame kernels which will consume some reactants before extinguishing. The question arises as to how much of the reactants is consumed in this process and how much heat is released. This information is relevant to the assessment of ignitability of the remaining material and the likelihood of the hot, decaying kernel of gas igniting an adjacent layer of mixture at a different thermodynamic state. Experiments by different investgators, summarized in [992], indicate that the ratio of the chemical enthalpy release before extinguishment ( Hchem) to the ignition energy input (Eign) is of the order of 5  15. From the volume of the ignition kernel just before extinguishment, Hchem was estimated to be Hchem 4 pg (1 e ) P r3 12 P r3 x x 3( g  1) (23) where rx is the radius of the flame kernel just before visible extinction. The latter approximation in Eq. (23) is representative of hydrocarbonair mixtures (g 1.4, e << 1). While it seems plausible t h a t H chem/Eign might usually be in the range 5  15, it would also seem that very weak sources might produce smaller values and barely subcritical sources much larger values. This has not been tested experimentally; such experiments would probably be difficult to perform because rather precise measurements of Eign would be required. Numerical computations are not subject to this limitation but apparently no such study of this effect has been performed. Since no systematic study of the effect of Eign on Hchem/Eign has been reported, such calculations are presented here. The numerical code employed in these calculations is the same used in [9915]; this code includes detailed chemical, transport, and hydrodynamic submodels. As Eign was varied, t h e radius of the ignition source (rs) was adjusted such that the peak temperature at the ignition source was consistently about 3000K. A dilute, lean hydrogenoxygennitrogen mixture with H 2 :O2 : N2 = 1:1:10 a t 15 1 atm pressure and ambient temperature of 300K was studied. Results of these calculations are shown in Fig. 6. It can be seen that for very weak sources, Hchem/Eign approaches unity, indicating that no chemical energy is released. This is expected because, due to the steep gradients, the energy deposited in these small kernels is diffused away more rapidly than heat is generated through chemical reaction. For Eign approaching Emin, H chem/Eign diverges. This is to be expected, since for Eign Emin, H chem/Eign is infinite, i.e. the flame propagates indefinitely. These calculations show why the experimentallyobserved range of H chem/Eign is not large. The repeatability of Eign in an experiment and the accuracy of its measurement is probably no better than 20%, at least for spark ignition sources. For intermediate values of Eign/Emin, in the range 0.2 to 0.8, Hchem/Eign only increases from 8 to 22 for the example shown. Thus most observed values of H chem/Eign would be expected to fall in this range, and indeed this is found experimentally except in some special cases discussed in section 8. 4.2 Transition from ignition to steady propagation It is useful to consider the ignition process as consisting of three stages, which overlap to some extent: 1) nonreacting ignition kernel evolution, 2) ignition kernel evolution supported by heat release from chemical reaction, and 3) development of steady flame propagation. While the model of ignition for Le < 1 described in section 3.2 may describe the initiation of t h e flame kernel for Le < 1, it does not describe the development of a steady flame because it is valid only on the small scale where Rf is comparable to Rz , which is much smaller than d when Le < 1 (see Eq. 17). Development of a steady flame can occur only for Rf >> d. Buckmaster [9942] described the transition of Zeldovich flames to steady propagating flames for Le < 1 using AEA. This theory suggests that Rf ~ t ln(t) during the transition. Of course, when steady propagation is achieved, Rf ~ t. When Le is close to or larger than unity, this transition from Zeldovich flames to steady flames does not apply at a l l because in this case Rz d and in any case it does not apply when R f d . However, numerical integrations of an evolution equation for spherical flame kernels [9961] with Le < 1 on the much larger scale Rf ~ bd show behavior suggesting Rf ~ reported t , which would be very similar to t ln(t) behavior. Moreover, some computatons based on primitivevariable formutions [9943] and experiments [992] have t behavior at Rf > d for lean CH4 air mixtures with Le 0.96 and thus Rz d. Thus, there is t transition may be expected over a wide substantial evidence that for mixtures with Le < 1, an R f ~ range of Rf spanning d. Thus dRf/dt approaches its steady value from larger values, and thus the total time and propagation distance needed to reach steadystate is less than if dRf/dt were constant. This point may be significant because it could substantially affect estimates of the time required for complete consumption of a given volume of combustible mixture in, for example, and IC engine. 16 For Le > 1, flame evolution is quite different. Numerical simulations [9943, 9911] predict that in very lean nearlimit C3 H 8 air flames flame kernels expand rapidly upon introduction of the ignition energy, decelerate to nearly zero propagation rate, then suddenly accelerate to the steady burning velocity. An example of this is shown in Figure 7. Champion et al. were barely able to observe t h e effect experimentally at oneg in a slightly lean C3 H 8 air flame, Because of the uncertainty in the spark energy measurement and the stochastic nature of t h e ignition process. This stochastic behavior has led some investigators [9924] to report ignition probability for a given measured spark energy rather than a definite value of MIE. 5.0 EFFECTS OF THE STATE OF THE COMBUSTIBLE MIXTURE One of the most important questions concerning ignition is how does the state of the material affect its ignitability. For example, T and P vary with compression, and altitute. What about lean or rich. etc. 5.1 Pressure Additionally the pressure effect discusses therein depends on S L being independent of pressure. This is reasonable for most hydrocarbons near stoich but not for near limit mixtures [9937]. For nearlimit mixtures of hydrocarbons in air, the order of reaction is <<2 and thus the effect of S L isn't t h e same. In fact, for very lean HC's Emin may increase with increasing pressure (figure***) [992,9944] 5.3 Stoichiometry The aforementioned analyses have all considered the case of a "one reactant" flame, i.e. a mixture far removed from the stoichiometric composition so that depletion of the abundant reactant is negligible. It could also represent (for example) an ozone decomposition flame. Clearly, when t h e composition is near stoichiometric this assumption is invalid. In this section effects of stoichiometry are discussed. Lewis and von Elbe [991] point out that the minima of Emin for hydrocarbons in air are nearly the same but shift to rich stoichiometries for higher molecular weight fuels. For since methane has a higher diffusivity than oxygen, but all other alkanes have lower diffusivity than oxygen, it is expected that the lowest Emin will occur on the lean side for methane (so that the fuel is the scarce reactant, and the Lewis number will be that based on methane) but on the rich side for other hydrocarbons (so that oxygen is the scarce reactant, and the Lewis number will be that based on 17 oxygen). Another factor is that for higher alkanes, add a little fuel and get lower alpha since its much heavier than nitrogen  helps both alpha directly and Le too. Also  check Wang and sibulkin Favored when the stoichiometrically deficient reactant has the lower Le, which is plausible based on the above discussions. Joulin [9913] further shows that Emin can change rapidly with stoichiometry because the deficient reactant at the flame front is not necessarily the globally deficient one; the transition from lean burning to rich burning occurs not at equivalence ratio () of unity but rather at c = Lefoel/LeO2. As a result, when crosses the special value c, R z may change rapidly, resulting in large changes in Emin ~ Rz 3 . One difficulty with prior experiments that has led to some confusion in the importance of t h e unequal rates of diffusion of thermal energy and mass versus the unequal rates of diffusion of reactants of differing diffusivity is that hydrocarbons in air have been used in most experiments. It is unfortunate in this context that aN2 DO2N2. Hence, the Lewis number of rich hydrocarbonair mixtures, i.e. a N2/DO2N2 by the standard definition, is nearly unity. Adding large amounts of a heavy hydrocarbon decreases aN2 more than DO2N2, hence the Lewis number decreases slightly. Thus it is difficult to distinguish between the between the preferential diffusion effect and the diffusivethermal effect by studying hydrocarbonair mixtures. Better to choose HeO2 or O2 CO2 "airs" 5.4 Additives effect of a light additive  Joulin's work  deserved further study for ign of lean mxitures When t h e overall mixture strength is adjusted to maintain a fixed Tad, Rz Rz( y B =0 ) b 1 1 ^^ hB y B ~~ ; D = exp D hB y B + hA y A 2 LeA LeB where Rz(yB=0) is the Zeldovich radius for the mixture without the light additive, h is the heating value, and the subscripts A and B refer to the base fuel and additive, respectively. Experiments show this for small H2 added to C3H8 [9911], but effect stronger than predicted, maybe due to difference in effective Ea for H2 vs. C3H8  H2 probably lower. 6.0 EFFECTS OF THE CHARACTERISTICS OF THE ENERGY DEPOSITION SOURCE For a given combustible mixture need to know how the ignition source affects emin 6.1 Volume 18 Experiments [991] show that the spark gap (ds) affects Emin. An optimum gap ds,opt which minimizes Emin is found; Emin increases slowly for ds > ds,opt and more rapidly for ds < ds,opt. For ds > ds,opt, the increase is readily comprehensible; if ds is significantly greater than R z for t h e mixture, a larger volume of mixture than required is being heated, and as would be expected in this regime, Emin ~ ds3 [991]. For ds < ds,opt, greater heat loss to the spark electrodes will result in a larger portion of the deposited energy being dissipated in ways which are not useful for ignition and/or the greater strength of the pressure waves (since the energy is being deposited in a smaller volume. In analytical and computational studies, the experimentally observed behavior for large ds is reproduced [994, 9914] as expected. However, most of these studies ([9915] being an exception) do not predict an increase in Emin at small ds but instead show Emin essentially constant for sufficiently small ds. In fact, analytical work [997, 998] indicates that the spatial distribution of energy distribution at radii smaller than Rz does not play any role. In any case, it is likely that most of the observed increase in Emin a t small ds is related to experimental difficulties in minimizing losses. Add F&S 3d figure While the potential effect of shock losses on Emin is well known, quantitative estimates have not been reported previously. According to the classical Taylor blast wave model [9949] of the response of a gas with specific heat ratio 1.4 to an instantaneous deposition of energy in a very small volume, t h e energy left behind in the form of internal energy of the gas is close to 1/3 of the initial energy deposition. Thus about 2/3 of the total spark energy could be lost to shock losses if the energy Also, because the extremely high deposition volume were very small. This is only an upper bound on the losses, however, because in experiments the energy deposition radius is not infinitesimal. temperatures generated by the laser spark will lead to dissociation of the gas, some energy will be left behind in the form of dissociated molecules that will liberate thermal energy as they recombine. Thus, it would seem that no more than perhaps 50% of the spark energy could be lost to shock losses. W h i l e this loss may be substantial, it cannot by itself account for the discrepancies between different ignition sources or the discrepancies between the the experiments and numerical calculations assuming isobaric heat input [9940]. Further evidence for this assertion can be found in the work by Frendi and Sibulkin [994] who employed an ignition model employing detailed gasdynamics but simplified onestep chemistry without dissociation and transport properties and found that the constantpressure Emin was about a factor of two lower than that computed when shockwave effects were included. 6.2 Duration Experiments have also shown that Emin is a function of the duration ( ts) of the energy input [9916] and that there is a t s which minimizes Emin. Clearly for t s sufficiently large, the flame 19 propagates away from the ignition source before the end of the deposition period, and Emin increases. Under these conditions, we expect Emin ~ ts, which is consistent with experimental [9916] and computational [994,998] results. The explanations for the increase in Emin for small ts require more thought. In experiments, this effect could be attributed to greater losses from the stronger pressure waves generated by the spark when ts is small, and this result is found in models which include such effects [994]. However, even theory and computations which assume constant pressure show this effect to some extent, and this has not previously received an interpretation. It is speculated here that this is a result of the difference between the temperature profiles of the stationary solution, in which T ~ 1/r in the far field, and t h e thermal profile resulting from a deltafunction input of energy, which is an error function. If t h e duration is longer it is more like a 1/r profile (since that's the steady result) but of course if its too long thats no good either. So you want it to be like a 1/r profile, which means long duration, but not so long that the flame has moved away. 6.3 Method of energy deposition 6.3.1 Spark discharges Breakdowntype spark discharges exhibit lower values of Emin than arctype discharges, and these in turn are more effective than glow discharges [9919, 9920]. This is primarily a function of the conduction losses to the spark electrodes, which varies for each type of discharge depending on its duration and distribution of energy in the plasma channel [9919]. 6.3.2 Thermal energy versus radicals It is informative to know whether sparks and other hightemperature sources ignite flames by deposition of active species or thermally. Most practical ignition sources are a combination of thermal and chemical sources. Different types of sparks give different results. An experiment would be difficult to do, but computationally not a problem. It has been shown that depositing energy in the form of dissociated reactants results in slightly lower Emin [9917, 9918]. Ignition delay time much lower because radical buildup time not needed when they are directly deposited. difference expected. Difference disappeared if T > 3000K for thermal source. 6.3.3 Lasers Would expect that an overabundance of an active species would quickly recombine and give off thermal energy so not much 20 A comparison of laser and conventional ignition sources has been published elsewhere [9946], hence only a brief summary will be given here. Laser ignition sources yield values of Emin comparable to sparks [9921], except at low pressure and for lean mixtures, where the laser yields lower Emin [9922]. This is attributed to the greater heat losses to spark electrodes under these conditions [9922]. 7.0 EFFECT OF FLOW ENVIRONMENT In most practical flames, ignition occurs in the presence of flow. For example, in internal combustion and turbojet engines, the flow in the combustion chamber is highly turbulent. Also, even in unintentional igntions the energy deposition will occur concurrently with buoyancyinduced flow. In this section these are considered. 7.1 Mean flow and mean strain It is also found that the flow environment affects Emin. Emin generally increases with increasing flow velocity (U) relative to the spark electrodes [9923, 9924], although U >> SL is required to see a significant effect. Consistent with the discussions above, the increase in Emin is attributed to t h e increase in developing flame kernel area caused by aerodynamic straining of the kernel [9923, 9924]. A more quantitative model of the effect of straining on Emin [9925] yields similar results. There is some evidence that low flow velocities [9924] or weak turbulence [9923] may decrease emin slightly. The models [99239925] do not predict this, although the effect is plausible because in practice the flow will positively strain the leading edge of the kernel and negatively strain t h e trailing edge. 7.2 Turbulence Turbulence is also found to increase Emin [9923]. On the other hand, there is some evidence t h a t low flow velocities [9924] or weak turbulence [9923] may decrease Emin slightly compared to stagnant mixtures [9924]. Ishii et al [9963] also show this. shock wave reduces turbulence but not clear why. The also found that longer sparks (more noncapacitive component) can ignte easier when mixtures is turbulent  not clear why. They suggest spark 9.0 IGNITION OF HETEROGENEOUS MIXTURES 9.1 Aerosols 21 Dilute aerosol sprays are found to exhibit an optimal droplet diameter which minimizes Emin [9927] and at the optimum, are comparable to Emin of stoichiometric gaseous mixtures. This result was attributed to the production of mixtures surrounding the spark which are locally very rich when t h e droplet size is small (since they evaporate readily due to their small size and high mobility) whereas large droplets produce locally very lean mixtures. Dust suspensions are found to have much higher values of Emin [9928], which was hypothesized to result from the high flow velocities and turbulence levels needed to suspend the dust. 9.3 Nonpremixed systems While all of the studies reported above considered premixed reactants, Schmieder [9947] studied the ignition of a nonpremixed flame, consisting of a CH4 gas jet issuing into ambient air, using CO2 laser sparks. Because the results were not consistent from shot to shot, Schmieder reported t h e probability of ignition as a function of position along the jet axis for fixed pulse energy rather than a definite value of Ef. He also found it was possible to extinguish flames with the laser spark. This was thought to be due to the strong shock wave caused by the spark which blew out the flame in much t h e same way as one blows out a candle. For some conditions the probability of a given laser spark blowing out a lighted flame was almost identical to the probability of the same spark igniting a nonreacting gas jet. Nonpremixed systems are those in which the reactant do not come together until the time of combustion. Not much work on the ignition of nonpremixed systems by sparks or other localized sources of ignition  much more on how one flame will ignite another  this is really flame spread or flashover and outside the scope of our work. Same with ignition of droplets by hot gas, e.g. in diesel. Beyond our scope. 10.0 PRACTICAL APPLICATIONS The problem of ignition also has many applications. An important example of this would be fire safety in oil refineries and coal mine shafts; much of the early work on ignition phenomena in gases was performed at the U.S. Bureau of Mines. This work is summarized in Chapter 5 of the classic text by Lewis and von Elbe [19]. Obviously an understanding the characteristics of flame ignition is essential to the development of proper fire safety techniques in these cases. Other applications would include t h e ignition of fuel/air mixtures in internal combustion engines and the highaltitude restarting of turbojet engines. The former is particularly important in light of the desire to operate with very lean mixtures, 22 as discussed above, for which values of Emin are much higher than those of nearstoichiometric mixtures, particular for heavy fuels (high lewis number). 11.0 CONCLUSIONS A simple diffusivethermal model is adequate to qualitatively describe the processes of spark ignition of combustible mixtures when Lewis number effects are included. Many effects of ignition source, flow environment, and mixture properties can be accounted for when thermal loss mechanisms are considered. Simple thermal models of ignition of deflagrations and detonations are adequate to describe qualitatively many of the processes of conventional and laser ignition of combustible mixtures, including the effects of the ignition source characteristics, mixture properties, and the flow environment. The energy requirements are generally not substantially different between conventional and laser ignition systems, and for deflagrations the energy requirements are quite modest. The primary benefits of laser ignition for practical combustion devices probably lie in the ability to choose the location(s) and timing of ignition events in ways which are not feasible with conventional ignition systems. These benefits may also facilitate the possibility of monitoring and controlling the burning process in real time. Finally, it should be noted that, with one exception, only the ignition of premixed reactant streams has been discussed here. In many applications (e.g. Aerospace Plane) may be mixing limited not combustion limited. The mixing which is necessary before combustion takes place cannot be enhanced by the ignition process. However, even if a combustion system requires a large mixing section upstream of the combustion section, there may still be some advantages of laser ignition sources. Since a proper ignition system may be able to reduce the overall burning time, it would be possible to burn quickly and expand the product gases to low temperatures. This would minimize the amount of NOx formation (since it is highly sensitive to temperature) and the shorter length of the combustion section indicates a reduced need for hightemperature material. REFERENCES 23 991. 992. 993. 994. 995. 996. 997. 998. 999. 9910. 9911. 9912. 9913. 9914. 9915. 9916. 9917. 9918. 9919. 9920. 9921. 9922. 9923. 9924. 9925. Lewis, B., von Elbe, G.: Combustion, Flames, and Explosions of Gases, 3rd ed., Academic Press, 1987. Ronney, P. D., Combust. Flame 62, 120 (1985). Williams, F. A., Combustion Theory, 2nd ed., BenjaminCummins, 1985. Frendi, A., Sibulkin, M.: "Dependence of Minimum Ignition Energy on Ignition Parameters," Combust. Sci. Tech. 73, 395413, 1990. Egolfopoulos, F., Law, C. K.: 23rd symposium. Zeldovich, Ya. B., Theory of Combustion and Detonation of Gases, Academy of Sciences (USSR), Moscow, 1944. Buckmaster, J., Weeratunga, S., Combust. Sci. Tech. 35, 287 (1984). Deshaies, B., Joulin, G., Combust. Sci. Tech. 37, 99 (1984). Ronney. P. D., Combust. Sci. Tech. 59, 123 (1988). Joulin, G.: Combust. Sci. Tech. 43, 99 (1985). Champion, M., Deshaies, B., Joulin, G., Kinoshita, K.: Combust. Flame 65, 319 (1986). Champion, M., Deshaies, B., Joulin, G.: Combust. Flame 74, 161 (1988). Joulin, G.: Siam J. Appl. Math. 47, 998 (1987). Maas, U., Warnatz, J., Combust. Flame 74:53 (1988). Kailasanath, K., Oran, E., Boris, J.: Combust. Flame 47, 173 (1982). Kono, M., Kumagai, S., Sakai, T.: 16th Symposium (International) on Combustion, Combustion Institute, 1976, p. 757. DixonLewis, G., Shepard, I. G.: 15th Symposium (International) on Combustion, Combustion Institute, 1974, p. 1483. Sloane, T. M.: Combust. Sci. Tech. 73, 351 (1990) Maly, R., Vogel, M.: 17th Symposium (International) on Combustion, Combustion Institute, 1978, p. 821. Ziegler, G. F. W., Wagner, E. P., Maly, R.: 20th Symposium (International) on Combustion, Combustion Institute, 1984, p. 1817. Kingdon, R. G., Weinberg, F. J.: 16th Symposium (International) on Combustion, Combustion Institute, 1976, p. 747. Weinberg, F. J., Wilson, J. R.: Proc. Roy. Soc. (London) A324, 41 (1971). Ballal, D. R., Lefebvre, A. H.: 15th Symposium (International) on Combustion, Combustion Institute, 1974, p. 1473. Kono, M., Hatori, K., Iinuma, K.: 20th Symposium (International) on Combustion, Combustion Institute, 1984, p. 133. Tromans, P. S., Furzeland, R. M.: 21st Symposium (International) on Combustion, Combustion Institute, 1986, p. 1891. 24 9926. 9927. De Soete, G. G.: 20th Symposium (International) on Combustion, Combustion Institute, 1984, p. 161. Singh, A. K., Polymeropoulous, C. E.: 21st Symposium (International) on Combustion, Combustion Institute, 1986, p. 513. 9929. Ishii, K., Tsukamoto, T., Ujiie, Y., Kono, M.: Combust. Flame 91, 153 (1992). 9930 chemical ignition 9931 flammability limits 9932 J. H. S. Lee, "Initiation of Gaseous Detonation," Ann. Rev. Phys. Chem, Vol. 28, pp. 75104, 1977; 9933 J. H. S. Lee, "Dynamic Parameters of Gaseous Detonations," Ann. Rev. Fluid Mech, Vol. 16, pp. 311336, 1984. 9934. R. A. Hill, "IgnitionDelay Times in LaserInitiated Combustion," Appl. Opt., Vol. 20, pp. 22392242, 1981. 9935. W. M. Trott, "CO2 laserinduced Deflagration of Fuel/oxygen Mixtures," J. Appl. Phys., Vol. 54, pp. 118130, 1983. 9936. M. Lavid and J. G. Stevens, "Photochemical Ignition of Premixed HydrogenOxidizer Mixtures with Excimer Lasers," Combust. Flame, Vol. 60, pp. 195202, 1985. 9937 reaction orders close to 2 for stoich but not lean 9938 bush and fendell 9939 Mitani and Williams 9940 Sloane and Ronney 9942 Buckmaster SEF root t paper 9943 farmer and ronney 9944 Polymeropoulos near limit mies 9945 swett T effect on Mie 9946 mushroom shaped flame 9947. R. W. Schmieder, "Laser Spark Ignition and Extinction of a Methaneair Diffusion Flame," J. Appl. Phys., Vol. 52, pp. 30003003, 1981. 9948 coffee 1 step methane 9949 Taylor, G. I. (1950). Proc. Roy. Soc. (London) Ser. A 201, 175. 9950 chinese paper 9951 My 1st ISC paper 25 9952 wang & sibulkin 9953 westbrook & dryer 9954 jones and lindstedt 9955 sloane's chemical mechanism 9956 ignition expt showing torus (other than Kono or Joulin) 9957 clavin pecs review 9958 my C&F flame balls paper 9959 joulin & sivashinsky cell effect on flammability 9960 Gatto AIAA paper 9961 ronney and sivashinsky 9962 buckmaster joulin ronney #1 9963 Ishii, K., Aoki, O., Ujiie, Y., Kono, M., 24th Symposium (International) on Combustion, Combustion Institute, 1992, p. ***. 9964 Kravchik, T., Sher, E., Combust. Flame 99, 635 (1994). 26 Figure Captions TEMPERATURE Unsuccessful ignition Initial profile Later Still later DISTANCE Successful ignition Initial profile Later Still later SL ATUREE T RE P M DISTANCE figure 1 schematic of ignition processes: a) unsuccessful b) successful 27 Minimum ignition energy (mJ) 10 10 10 10 10 10 2 Hydrogen (lean) Hydrogen (rich) Methane (lean) Methane (rich) Ethane (lean) Ethane (rich) Propane (lean) Propane (rich) Best fit to all data Slope = 0.739 1 0 1 2 Slope = 1 3 10 6 10 5 10 4 10 3 10 2 10
3 1 10 0 3 Pressure * (quenching distance) (atm cm ) Figure 2. Effect of quenching distance and pressure on minimum ignition energy, correlated according to scaling predicted by Eq. (5). Data points taken from Lewis and von Elbe [1]. 28 1000 100
3 3 R z / R z (Le = 1) e = 1/7 b = 10 10 1 0.1 0.01 0.001 0 0.5 1 Lewis number 1.5 2 Figure 3. Effect of Lewis number on the critical ignition kernel volume (Rz 3 ) figure 4 Joulin's R vs. T figure 5 Kono sequence, with comparison to calculations 29 40 H 2 : O2 : N2 = 1:1:10 30
Echem /Espark 20 10 0 0 0.2 0.4 0.6 Espark/Emin 0.8 1 Figure 6. Dynamics of subcritical ignition kernels: effect of spark energy (Espark) on energy liberated due to chemical reaction (Echem/Espark) before kernel extinguishes. Fig 7 S vs. t for lean propane 30 Reference Lewis & von Elbe [991] Ronney [992] This work, Eq. (4) Chinese paper [9950] Frendi & Sibulkin [994] Wang & Sibulkin [9952] Tromans & Furzeland [9925] Sloane & Ronney [9940] Sloane & Ronney [9940] Sloane & Ronney [9940] Kono [***] Type of study Experiment Experiment Phenomenological model Computation, 1step chemistry Computation, 1step chemistry [9948] Computation, 4step chemistry [9954] Computation, 1step chemistry [***] Computation, 1step chemistry [9948] Computation, 1step chemistry [9953] Computation, detailed chemistry (A) 0.33 0.5 0.011 (B) 0.41 0.010 0.23 0.005 0.012 0.012 0.010 0.012 0.10 small Table 1. Comparison of measured and predicted values of minimum ignition energies for (A) stoichiometric CH4 air mixtures at 1 atm and (B) stoichiometric C3 H 8 air mixtures at 1 atm 31 ...
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