2534
J. A. Miller and R. J. Kee
3. In this case, the initial conditions were essentially the same as in previous runs. However, as fuel was burned, more fuel was injected (hot but with zero velocity) into the bottom center of the test chamber. As can be seen, the fuel oxidizes as it rises and mixes with the oxygen. Figure 4 shows the actual computer printout from a test similar to that shown in Figure 3. It should be noted that the contour levels are not constant from frame to frame, but are chosen to show the position of each of the species. Conclusions We are developing a two-dimensional hydrocode with self-consistant transport and chemical kinetics. We use
a two-dimensional model, since we are interested in the problems of mixing and turbulence, which cannot be done in a satisfactory manner using one-point calculations, or one-dimensional models. While the results of our simple tests so far are extremely encouraging, considerable careful and quantitative analysis of the results is called for and detailed diagnostic routines have to be developed before more interesting and intricate problems can be attacked.
References and Notes (1) D. L. Book, J. P. Boris, and K. Hain, J . Comp. Phys., 18, 248 (1975). (2) F.W. Sears, "An Introduction to Thermodynamics,The Kinetic Theory of Gases and Statistical Mechanics",Addison-Wesley, Reading, Mass., 1964; T. R. Marrero and E. A. Mason, J . Phys. Chem. Ref. Data, 1, 3 (1972).
Chemical Nonequilibrium Effects in Hydrogen-Air Laminar Jet Diffusion Flames James A. Miller" and Robert J. Kee Combustion Research Division 835 1, Sandia Laboratories, Livermore, California 94550 (Received April 26, 1977) Publication costs assisted by Sandia Laboratories
A theoretical model for the structure of hydrogen-air laminar jet diffusion flames has been developed. The model includes the effects of nonequilibrium chemical kinetics and realistic transport properties. The reaction zone in such flames is found to resemble the recombination zone in premixed flames. Distributions of the free radicals H, 0, and OH are substantially in excess of local equilibrium values. The computed results show that the partial equilibrium approximation, which is often employed in premixed flames, is valid within the reaction zone but breaks down outside it. Hydroperoxyl is found to be an important intermediate species in the flame, being formed primarily by the recombination reaction H + O2 + M HOz + M and destroyed by H, 0, and OH radical attack. The absence of chemical equilibrium and the broadening of the energy release zone result in peak temperatures which are well below the adiabatic flame temperature. Broadening of the energy release zone is due significantly to the mobility of hydrogen atoms.
Introduction The purpose of the present investigation is to examine, by means of a theoretical model, the effects of nonequilibrium chemical kinetics on the structure of H2/air laminar jet diffusion flames. Such a detailed treatment has not appeared previously in the literature because of the inherent difficulties in handling the multiple, vastly differing, time and length scales characteristic of the problem. These disparate scales make the integration of the differential equations governing the behavior of the flame a troublesome task. Historically, it has been necessary to resort to simplifying assumptions about the chemical kinetics in order to obtain solutions. To illustrate the point and to define the problem more precisely, consider a fuel jet which is issuing vertically into ambient still air and that has been ignited for a sufficient time to allow a steady flow to develop (Figure 1). We can think of the flame length L as the characteristic convective length scale in the problem and the nozzle radius R as characteristic of the diffusive scales. The usual situation in diffusion flames is that the nozzle radius is small compared to the flame length. In terms of time scales, we define Td = R / V d and 7, = L / U , where U and v d are characteristic convective and radial diffusive velocities, respectively. The nature of the physical problem is such that Td and 7, are of the same order of magnitude (Le., diffusion of mass and energy must occur between r = R and r = 0 over the axial length L in order to burn all the The Journal of Physical Chemistry. Voi. 8 1, No. 25, 1977
fuel), and we may think of one time as being characteristic of the fluid and transport aspects of the problem. A third time scale of interest is TQ, the time associated with the chemical conversion of reactants into products and the accompanying evolution of chemical energy. At high temperatures, where chemical reactions occur very rapidly, TQ typically is very small compared to Td or 7,; that is, the Damkohler number is very large, Td/TQ >> 1,and intense chemical activity might be expected to confine itself to a relatively narrow region between the ambient air and the fuel stream. To be specific, Td/TQ = 103-104for problems of interest here. The problem is complicated further, and most importantly, by the fact that TQ is only one of the largest of a number of time scales associated with the complex chemical reaction. Times associated with the creation and destruction of reaction intermediates, particularly free radicals, typically are much smaller than r~ These disparate time scales lie at the heart of the computational problems in predicting the structure of laminar jet diffusion flames. Historically, diffusion flame problems have been solved and by taking one or both of the two limits T d / r Q Td/TQ 0. The first limit constitutes equilibrium flow and the second, frozen flow. Both eliminate the need for the explicit consideration of any of the chemical time scales. The Burke-Schumannl flame sheet approximation effectively results in infinitely fast reactions in the reaction zone, reducing this zone to a flame sheet. Frozen flow then
-
-
2535
Hydrogen-Air Laminar Jet Diffusion Flames
in cylindrical coordinates as Continuity
apu 1 a - + --(prv) ax r ar
I
=
0
axial momentum
L
thermal energy
1
species
equation-of-state
FUEL
Flgure 1. Schematic of a laminar jet diffusion flame. Flame tip is defined specifically to be the point of peak centerline temperature.
results in the rest of the flame. Most solutions to the diffusion flame problem have followed this Libby and Economos6 have extended the concept somewhat by allowing for a finite zone within which rd/r$ m. At least one investigation' has assumed infinitely fast reaction everywhere. All of these approaches, however, preclude a detailed examination of the interplay between convective, diffusive, and chemical processes in the flame. The purpose of this paper is to provide such an examination for an H2-02-N2 reaction system. Williams8 recently has reviewed progress in the theory of diffusion flames of all types. Solutions to related problems, which involve chemical reactions in shear layers, stagnation point boundary layers, and flat plate boundary layers, have appeared in the literat~re.~J'However, they either are analytical, and thus require an oversimplification of the physical problem, or they involve characteristic flow times (7, = L / U ) which are more nearly equal to the chemical times than is the case for typical jet diffusion flames. In the present paper we describe the results of our calculations for a hydrogen flame burning in air, concentrating on chemical mechanism and the effects of reaction intermediates. In a companion paper12 we describe in detail the numerical solution technique which we have used to obtain such results.
-
The Mathematical Model The Governing Equations. By considering flames in which axial convective fluxes are much larger than diffusive fluxes, we may replace the Navier-Stokes equations by the boundary layer equations for a multicomponent gas. Such flames are characterized by a flame length which is much larger than the nozzle radius. The radial momentum equation is replaced by the condition that p = p ( x ) only, with p ( x ) imposed by the ambient air condition. The assumption of a thermally perfect gas is also imposed. The governing differential equations, then, are written below
In these equations x and r represent axial and radial coordinates respectively; u and u, the axial and radial fluid velocities; p , the mass density; p , the thermodynamic pressure; T , the temperature; g , the acceleration of gravity; Yk,the mass fraction of the kth species; wk, the molecular weight; a,the universal gas constant; D k - N P , the binary diffusion coefficient of the kth species into nitrogen; p, the viscosity; A, the conductivity; w k , the rate of production of the kth species by chemical reaction. The subscript e refers to the ambient air condition. In eq 2 the pressure gradient term and the gravitational term have been combined to give the buoyancy force ( p e - p)g. Otherwise, the pressure is assumed to be uniform. In formulating eq 4 we have recognized that molecular nitrogen is always the dominant species in the reaction zone, and since binary diffusion coefficients Dk, do not vary 1, VN, greatly with different species j , the limit XN, 0 has been taken in the Stefan-Maxwell equations, where x is the mole fraction and V is the diffusion velocity. This gives
- -
12v
In making such an approximation it is important to ensure that the diffusion velocities (and diffusion coefficients) defined by (6) are consistent with conservation of mass. This is accomplished by replacing the N2species equation by the condition that the sum of the mass fractions equals unity. N k= 1
y,=l
(7)
It is not necessary then to specify a diffusion coefficient for nitrogen, the consistent one being taken automatically. Also, in the equations introduced above, thermal diffusion and viscous dissipation have been neglected. Equations 1, 2, 3, 4,5, and 7 form a set of N + 4 equations for the dependent variables u , u , T , p , and Yk. Thermodynamic and Transport Properties. All transport properties for individual species have been The Journal of Physical Chemlstty, Vol. 8 1, No. 25, 1977
2536
J. A. Miller and R. J. Kee
TABLE I: Reaction Mechanism for H,-Air Combustion AH"
Reaction H, + 0, Z 2 OH H, + O H 8 H,O + H H + 0,sO H + 0 0 t H,@ OH + H H + 0, t M , i HO, t M OH t HO,.2 H,O t 0 , H t HO, 2 2 OH 0 t H O , g 0, + OH OH + O H s 0 + H,O H, t M s H + H + M #'H,O M = H,O 11. 0 , + M * 0 + 0 t M 12. H + O H + M g H , O + M M f H,O M = H,O 13. 0 t N , s NO + N 14. N t O , P NO t 0 15. OH + N s NO + H 16. H + H O , e H, t 0, 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. M
a &,=
Rate constanta (mol. cm3. s) kf, =1.7 x l o 1 , exp(-4778O/&T) k f , = 5.2 x l O I 3 exp(-6500/&,T) k f , = 1.22 x 1 0 1 7 T - 0 ~ 9exp(0 7 16620/6?,T) kf, = 1.8x 10'T exp(-8826/&T) k f 5= 2.00 X 1015exp(870/6?,T) kf, = 1.2 x 1013 kf,= 6 x l o L 3 kfg= 1 x i o L 3 kf, = 1.7 x 106T2.03 exp(llSO/&,T)
16.7 1.9 -45.9 -72.1 -38.7 -55.4 - 16.7 103.2
16 17 18 19 20 21 22 22,23 24 25
-118.0
26 27
75.0 -32.0 -48.7 -57.3
28 28 29 22
118.0
k f : , = 1.4 X l O I 4 exp(-75800/&,T) kf14= 6.4 X 109Texp(- 6280/&T) k f , , = 4.0 x l o ' ' k f 1 6 = 1.3 x 1013 -1"
1.987 cal/mol K.
where X kis the chemical symbol for the hth species and the sum is over all species being considered. The rate of progress of the ith reaction 6i is defined as the forward rate minus the reverse rate for the reaction and is given by si=
k
N
fl fi k=l
(9)
where
1 (1 + -); a
-"*E
+ ( 3 1 ' 2 ( 3 1 ' 4 ]
(
10)
Thermodynamic properties (heat capacities, entropies, and enthalpies) are computed from the JANNAF data used in the NASA Complex Chemical Equilibrium Code.15 Equilibrium constants in terms of partial pressures are obtained from the relation
Kpi = exp(- A4/62T + Agj62)
(11)
The corresponding equilibrium constant in terms of concentrations is given by N
where viki' and v,k are stoichiometric coefficients in the ith reaction (defined below). Chemical Kinetics. An elementary chemical reaction can be expressed in the form N
N 8
k=l
Vih"Xk
The Journal of Physical Chemistry, Vol. 81, No. 25, 1977
(13)
-
kr. fl 1
[&]'ik"
k=l
(15)
where Y,k = V,kf' - v,;. The reverse rates are given in terms of the forward rates and the equilibrium constants by
kr1. = k fi /K,
j= 1
N
[%lvik'
where [ x k ] denotes the concentration of the hth species. The species production terms in the conservation equations are found from 1
h -1
18.6
- 14.8
Ref
k f I 2= 7.5 x 1023T-2.6 kfl,%'= 20kf,,
;x = ?Vik&
z v*'%
kcal/mol
kf,,= 2.23 x 10IZT1'Z exp(-92600/&,T) k f l o H , O = 5kf10 k f , , = 1.85 x 1011T1'2 exp(-95560/&T)
calculated from the appropriate Enskog-Chapman expressions using the Lennard-Jones parameters given by S ~ e h 1 a . lCorrelations ~ of these properties as a function of temperature have been given by Pohl14and are used in the calculations presented here. The viscosity and conductivity of the gas mixture are obtained from the singlecomponent transport properties using Wilke's semiempirical formulas.
Qkj=-
(at
0 K),
(16)
The reaction mechanism we use for the oxidation of molecular hydrogen by air is given in Table I. Also listed are the rate constants used and the references from which they were taken or deduced. Because hydrogen peroxide was judged to be unimportant both as a stable product and as a reaction intermediate, reactions involving it were not included. However, reactions involving the h';idroperoxyl radical (HO,) were included to permit us to predict ignition limits accurately and to describe the recombination process properly. Careful consideration was given to the attack on HOz by the active intermediates H, 0, and OH. The role of HOz in the flame is discussed in detail in a later section. The nitric oxide formation is described by the extended Zeldovich mechanism. It was found desirable to include higher third-body efficiencies for water in some of the recombination reactions, as shown in the table. Boundary Conditions and Initial Conditions. The parabolic boundary layer equations require boundary conditions at r = 0 and r = re, where e denotes the outer edge of the jet, Initial conditions must be imposed at the burner exit plane. Boundary conditions are imposed so that r = 0 is a line of symmetry and so that at r = re, the dependent variables have values corresponding to ambient air. The appropriate boundary conditions are listed in eq 17. In order to accommodate the growth of the jet, the value of re is continuously adjusted as the solution is continued downstream; thus a solution that is both ac-
2537
Hydrogen-Air Laminar Jet Diffusion Flames
atr=O at r = re (au/ar)(O) = 0 u(r,) = u, (aT/ar)(O) = 0 T(r,) = T , (aYk/ar)(O) = 0
yk(re)
v(O)= 0
=
yke
curate and numerically efficient is maintained." Formulation of the initial conditions requires some care. The hydrogen and air are mixed over a narrow region in the vicinity of r = R and constrained so that H2 and 02 are in stoichiometric proportions at r = R. A Gaussian distribution centered at the stoichiometric point is imposed for the temperature. The width of the mixing zone and the height and width of the Gaussian temperature profile are somewhat arbitrary. Of course, the peak temperature must be above the ignition temperature (about 850 K) in order to ensure that the chain branching reactions take place, That is, the reaction H 02 P OH + 0 must dominate over H O2 M + HOz M. It is desireable to keep the widths of the mixing zone and temperature profile as small as is necessary to ensure that they do not unduly affect the solution downstream. Fortunately, we have found that transients die out very rapidly, and the solution is relatively insensitive to the initial conditions. Nevertheless, hydrogen on the lean side of the stoichiometric point and oxygen on the rich side react slowly and the effects of their presence persist downstream. We believe, however, that these initial conditions are physically reasonable (at least qualitatively), since some mixing of the fuel and oxidizer must take place in the vicinity of the nozzle lip prior to ignition. The axial velocity profile is taken to have the form of the similar solution for an incompressible, isothermal circular jet. This profile approximates the one that might be expected to emerge from the nozzle. A more detailed discussion of the implementation of initial and boundary conditions is contained in our paper on the numerical algorithms.12 T h e Solution Technique. Since ref 12 gives the details of our solution procedure, it is only necessary to sketch its salient features here. First, transformation is made from the r, x coordinate system to an 7, x system, where 7 = r / x . This procedure allows us to maintain the same number of grid points in the jet a t all axial stations, yet by simply adjusting the value of qe, to adjust continuously the position a t which the outer edge boundary condition is applied. Also, rather than solve the continuity equation in the form of eq 1,we use Krause's scheme30to eliminate from eq 1 the x derivatives by substituting the other conservation equations. This procedure produces an ordinary differential equation which is solved for the radial velocity, once the solution for u, T , and Yk have been obtained a t a particular axial location. The solution to the equations is obtained by an operator splitting technique of the majorant type, as discussed by Y a n e n k ~ .The ~ ~ axial convection terms in the transport equations are used as marching operators. Note that the other terms in the energy and species conservation equations are characterized by a time scale on the order of T, and T d (the convective and diffusive terms) or by time scales on the order of TQ or smaller (the chemical "source" terms). One continues the solution from one axial location to another by first using an implicit finite difference procedure to solve the equations without the chemical terms. The result is used as a predictor in the chemical portion of the split, which is formulated by considering only the marching operators and the chemical source terms. Since the chemical source terms contain no radial derivatives, a radially decoupled set of rate equations
+ +
+
150 CWS
LOCUSOF MAXIMUM TEMPERATURE POINTS
, ,
,
'
I
~
~
+
4 ISOTHER N S
CONSTANT VELOCITY CONTOURS
Flgure 2. Isotherms and velocity contours for a typical flame. Burner nozzle radius = 3.2 mm, centerline velocity = 600 cm/s (volume flow rate = 19 cm3/s).
results. These then are solved at each 9 node independently. This decoupling is an advantage of the scheme. Relatively large transport steps can be taken with many smaller chemical steps taken in the same interval. Since the chemical step sizes also are variable from node to node in the 7 direction, optimal efficiency and accuracy may be maintained. Thus the time scales in the problem are effectively decoupled. The Hindmar~h-Gear~~ algorithm is used to solve the chemical rate equations and has been found to be very effective in dealing with "stiffness" problems associated with very short free radical lifetimes (Le., very large negative eigenvalues of the Jacobian matrix). To increase the accuracy and efficiency of the overall method, chemical and transport operators are applied in reverse order a t alternate steps.
Results and Discussion Detailed mathematical models of flame structure serve two purposes. The first, and most obvious, is the detailed mapping of concentrations, temperatures, and velocities throughout the flame for comparison with experiment. The second, and perhaps more important, is the interpretation of such results in terms of simple conceptual models. In presenting the results of our calculations, our aim is to give a concise physical description of the processes which determine flame structure, using the numerical results principally to illustrate and support our conclusions. General Flame Characteristics. Figure 2 shows both isotherms and constant velocity contours for a typical flame. In this calculation, the radius of the burner nozzle is 3.2 mm, and the peak centerline velocity is 600 cm/s (volume flow rate = 19 cm3/s). The calculations reported in this investigation deal with flames which are characterized by Reynolds numbers p U ( 2 R ) l ~of the order lo2 and Froude numbers V / g L in the range 1-10. The jagged line in the isotherm plot of Figure 2 represents the locus of maximum temperature points in the radial profiles. Peak temperatures in the flame are less than 2200 K, even though the adiabatic flame temperature is 2379 K for stoichiometric hydrogen-air mixtures. These low temThe Journal of Physical Chemlstry, Vol. 81,
No. 25, 1977
J. A. Miller and R. J. Kee
2538
2200r
10
0-
L
1 400
01
0 2 3 5 04
05
0'6
07
08
09
1'0
11
o
o
A
peratures appear to be due not only to the use of realistic transport properties in the calculation (for example, the c ~ in the reaction zone) but Lewis number X l p D ~ is~ 0.3-0.4 also to deviations from chemical equilibrium in the reaction zone. It is particularly interesting to note the sharp axial temperature gradients outside the nozzle lip in the vicinity of the exit plane. This effect is due largely to the high hydrogen diffusion coefficient. In fact in some preliminary experimental work indications have been that the flame actually extends below the nozzle exit plane. Thus the boundary layer approximations, which are made in deriving the governing equations, are unquestionably deficient in this region. However, in the vast majority of the flame the boundary layer approximations do appear to be quite good. Due to the low fluid velocities, the chainbranching ignition process, typical of hydrogen-oxygen reactions, occurs only in a relatively narrow region near the lip of the burner nozzle (essentially confined well within the region where the isotherms bow out very sharply). Chemical reaction in the downstream portions of the flame resembles the recombination zones of premixed flames, except that the radical pool is constantly replenished by hydrogen and oxygen diffusing into the reaction zone and undergoing chemical transformation. Radial jet profiles reveal that a well-defined structure is maintained throughout the flame. Variations in the radial position of the reaction zone occur primarily because of slowly changing fluxes of hydrogen and oxygen to this zone. Near the flame tip, there is no longer sufficient hydrogen in the core of the jet to support the required flux of fuel to the reaction zone. As the fuel is depleted, then, the flame zone moves toward the centerline, and the temperatures drop as energy is diffused away. In the velocity contour plot, note the behavior of the 75 and 150 cm/s contours. These bow out away from the centerline as they pass through the high-temperature regions of the jet. This effect is due to buoyancy. As a fluid particle orginating in the fuel jet is heated, its density decreases and the buoyancy force begins to dominate the shear. The result is the acceleration of the fluid particle (streamlines in the interior of the jet are essentially parallel to the centerline). In fact, buoyancy is a very important factor in determining the behavior of flames in such low momentum jets. From the few calculations that we have done with zero gravity, it is apparent that, without the effect of gravity, flames bow out much farther than the one shown here and are somewhat longer. That is, the buoyancy force accelerates the flow in the reaction zone, causes shearing, and thus increases the air entrainment rate. As a result, the total oxygen flux to the reaction zone is greater in the buoyant case, enhanced markedly by the induced radial convective motion. This effect allows the flame to sit closer to the centerline and result8 in more rapid depletion of the fuel. Figure 3 shows radial profiles of the axial and radial convective velocities at an axial station (x = 1.07 cm) slightly less than half way between The Journal of Physical Chemistry, Vol. 81, No. 25, 1977
,
8
09
10
1 1
RADIUS ICMI
RADIUS ICh?
Figure 3. Axial and radial velocities at x = 1.07 cm.
o
Flgure 4. Major species and temperature profiles at x = 1.07 cm.
I
'0'
A H 2 ,
01
02
03
4
(EQUILIBRIUM1
i 04
05
06
07
08
09
10
11
RADIUS (CMI
Figure 5. H20 mole fraction profile compared to local chemical equilibrium. Significant deviations from equilibrium exist between r = 0.40 cm and r = 0.60 cm (x = 1.07 cm).
the nozzle exit plane and the flame tip. The figure reveals substantial axial velocities outside r = R (due to buoyancy) and substantial entrainment rates between r = 0.4 and 0.7 cm, where the temperatures are very high. Figure 4 is a plot of the major species profiles and the temperature at the same x = 1.07 cm axial location. Figure 5 shows the water mole fraction profile, compared to local chemical equilibrium. Equilibrium calculations are based on the local temperature and element composition. Obviously, nitrogen very quickly becomes the dominant species in the reaction zone and a major component in the central core of the jet. The hydrogen and oxygen maintain gradients in such a relationship that their total fluxes (diffusive and convective) to the reaction zone (i.e., the 2 mm thick region between r = 0.4 cm and r = 0.6 cm where water vapor is noticeably out of equilibrium) are approximately in stoichiometric proportions. This relationship is the major criterion governing the radial position of the reaction zone. As the hydrogen is depleted further downstream, the reaction zone moves in toward the centerline in order to try to maintain this condition. Figure 5 shows that the greatest deviation of the water concentration from its equilibrium value (i.e., the region of greatest chemical activity) occurs slightly to the lean side of the water concentration peak and on the outer edge of the peak temperature region. Noting Figure 3, one can see that as a fluid particle is convected across the reaction zone from outside to inside, chemical reactions occurring inside it produce water and raise the temperature at a finite rate (although not all the energy release is directly connected with H 2 0 formation). Consequently, the water concentration and temperature peaks are slightly displaced to the inside of the reaction zone center. Thus the radial convective velocity also serves to increase chemical activity by removing water from the reaction zone, resulting in a higher degree of chemical nonequilibrium than would otherwise occur. The absence of equilibrium water concentrations at radial positions outside r = 0.7 cm, as shown in Figure 5, is due to the very slow oxidation of hydrogen which originated on the lean side of the stoichiometric point at the nozzle lip. Later figures show some minor
2539
Hydrogen-Air Laminar Jet Diffusion Flames
1
0 008
:
PARTIAL
0005
~-
/ '0
20 30 40 50 60 VOLUME FLOW RATE (CM'ISECI
10
70
0'
80
Flgure 8. Linear correhtion of flame length with volume flow rate. Data points represent three flame calculations with R = 3.2 mm. The line shown is a linear representation of the calculated points. 02
01
0.2
0.3
0.4
OH
t
,
00024
E
H2 Z H 2 0 + H
.
I
1
1.1
I
I I
I
Ly
1
/ \ 1
1
,EQUILIBRIUM
(R121 H + O H t M % H Z O t M
02
01
03
04
05
06
07
OB
09
10
1
RADIUS iCM1
11
RADIUS (CMI
Flgure 7. Radial profile of the quantity b , A H ; / p for the most important energy-releasing reactions. Significant contributions come from reactions H H M F! H2 M (RlO),which forms the fuel, and H -I-O2 M e H02 M (R5) ( x = 1.07 cm).
+
, 10
~
9 l O H t O H Z OtHZO
+ +
, 0.9
'
ill 00016
:
=
1 OH t H 0 2 H 2 0 + O2 1 H t H02 OH ' O H
, 08
Figure 8. H atom mole fraction compared to local equilibrium and partial equilibrium ( x = 1.07 cm).
I i H + O Z t M r HOZ+M
-OO
\I,
0 5 0.6 0.7 RADIUS I C V I
+
Figure 9, 0 atom mole fraction compared to local equilibrium and partial equilibrium ( x = 1.07 cm).
+
chemical activity in this region, but the reaction takes place slowly. Flames in jets of different flow velocities behave essentially the same as the one described above. The flame length (defined as the distance from the nozzle exit plane to the flame tip) is the most apparent change. Figure 6 shows a plot of flame length vs. volume flow rate for flames computed with the same geometry as the one in Figure 2. The correlation is linear, as is expected, and should remain the same independent of nozzle radius. Chemical Mechanism i n the Reaction Zone. In the previous section we indicated that in the reaction zone there was a high degree of chemical nonequilibrium, as indicated by the difference between the H20concentration and its equilibrium value. This ratio typically deviates from one by as much as 15%. As the fuel in the jet core is depleted, however, and the reaction zone moves in toward the centerline, the water concentration begins to come to equilibrium. At the flame tip, water essentially has its equilibrium concentration throughout the cross section of the jet. In this section we want to examine more closely the details of the kinetic processes occurring in the nonequilibrium reaction zone. First consider Figure 7 , which is a radial profile of specific reaction energy release rates a t the x = 1.07 cm cross section. The function actually plotted is &AHLo/p, i.e., the energy release rate per unit mass. As one might expect, the most important energy releasing reaction is the three-body recombination reaction (R12),35which forms H + OH + M e H,O + M (R12) water. The rate of this reaction peaks, of course, where the water concentration deviates most strongly from its equilibrium value. The figure also reveals two other important and interesting results. First, hydrogen atom recombination is a dominant energy-releasing reaction in the flame. It occurs primarily on the rich side of the reaction zone, extending well into the core of the jet. Second, chemical reactions involving hydroperoxyl con-
I
0 W8;
4
0
;
'
i
0.0061
\OH
1
I \
OOO,!
0.002j
00
, p A R EQUILIBRIUM 0.1
0.2
0.3
T
d
0.4
A
0.5
l
a
R
0.6
0.7
I
u
1
M
0.8
0.9
1.0
1.1
RADIUS (CMI
Flgure 10. OH mole fraction compared to local equilibrium and partial equilibrium. Note that 0 and OH concentrations peak on the lean side of the reaction zone, allowing reaction R5 to dominate over R3 as a consumer of O2 ( x = 1.07 cm).
tribute significantly to the energy release. The effects of the H o p formation reaction H + 0 , + M* HO, + M (R5) and the exothermic destruction reactions OH + HO,@ H,O + 0, H + HO,* OH + OH
(R6) (R7)
combine to produce substantial energy release, primarily on the lean side of the reaction zone. One can see, then, that not all of the energy release is either directly connected with water formation or confined to a particularly narrow zone in the flame-contrary to what flame sheet concepts suggest. Both the absence of chemical equilibrium and the width of the energy release zone (also a manifestation of chemical nonequilibrium) contribute to the low peak temperature shown in Figure 4. The width of the energy release zone is strongly influenced by the mobility of the hydrogen atoms. All three of the most important energy-releasing processes involve H. The relatively small residence times of hydrogen atoms (high diffusion coefficient) allow them to be active chemically in all regions of the flame. Figures 8-10 show the radial profiles of H, 0, and OH mole fractions. In the central portion of the jet, the reThe Journal of Physlcal Chemistry, Vol. 8 1, No. 25, 1977
J. A. Miller and R. J. Kee
2540 03
0
- ___
-~
7
l R 5 1 H + O2 t M
~
H02
- hl
OH t HO,$ H,O t 0 ,
(Re)
H t HO,@ OH t OH
(R7)
0 t HO,+ 0 , t OH
(R8) (RIG)
H t HO,+ H, t 0,
020 0 01
i
02
03
04
_-
05
06
07
Ox
09
l
E
l
RADIUS ICMI
Flgure 11. Radial profile of viHOn8,/p for the reactions most important in creating and destroying H02. Note that H02 is essentially in steady state in the reaction zone, being destroyed as fast as it is created.
action zone concentrations of these species are about an order of magnitude above their equilibrium values. As distance from the reaction zone increases in the radial direction, the free radical concentrations drop, but the ratio of actual to equilibrium values gets larger. The chain branching ignition process, which creates the superequilibrium radical concentrations, is confined in the region directly adjacent to the nozzle lip. The superequilibrium condition in the reaction zone is maintained in the central portion of the jet by reactions involving H2 and 02,which diffuse into the reaction zone. The free radicals which occur in the rich and lean portions of the jet get there primarily by diffusion. Reaction in these regions is essentially frozen, and an even more pronounced superequilibrium condition exists there than in the reaction zone. Nonequilibrium free radical concentrations persist even past the flame tip. As the fuel burns out, these species are no longer constantly replenished and tend to recombine and form water. However, the temperature also drops and forces the local equilibrium concentrations down as well and, as a result, the free radical concentrations remain well above their local equilibrium values in the post-flame-tip region. A particularly interesting phenomenon occurs with the hydrogen atoms. Because of their low molecular weight, these atoms have high mobility and are able more easily than other species to diffuse away from the reaction zone where they are formed. Diffusion toward the lean region is obstructed by chemical reaction with oxygen. On the other hand, since diffusion into the rich zone is encumbered only by the relatively slow reaction R10, diffusion toward the centerline takes place readily. Hydrogen atoms thus build up to well over 1 X mole fraction on the centerline, recombination by reaction R10 occurs, energy is released, and molecular hydrogen actually has a net positive chemical formation rate in the rich zone. The process is actually enhanced slightly by the negative temperature dependence of the recombination reaction. The hydroperoxyl-forming reaction R5 is allowed to take place in the high-temperature regions of the flame for the following reason: Reaction R3 is sufficiently close to H + 0 ,* OH
+0
(R3)
equilibration in the reaction zone to allow (R5) to compete favorably. The factor which governs the competition between reactions R3 and R5 is the OH and 0 concentrations. Where these are sufficiently high, reaction R3 actually has a small net rate of progress in the reverse direction, allowing reaction R5 to dominate. As OH and 0 drop off toward equilibrium, however, reaction R3 can proceed in the forward direction and compete with reaction R5. Although HOz is formed a t a substantial rate in the reaction zone, it has a very short lifetime. Attack by H, 0,and OH through reactions R6, R7, R8, and R16 destroys The Journal of Physical Chemistry, Vol. 81, No. 25, 1977
it immediately upon creation. Figure 11,which is a radial profile of the quantity V , H O ~ & / P(the chemical production rates of H 0 2by the elementary reactions of interest), shows that HO2 is essentially in steady state with the net destruction rate approximately equal to the creation rate. Hydroperoxyl is an extremely important intermediate in the flame; reaction R5, which forms HOz, contributes significantly to the energy release and also represents the reaction step with the largest O2 consumption rate, as will be shown below. The process of H 0 2 production and destruction also helps significantly in replenishing OH in the reaction zone. The role of H 0 2 in premixed hydrogen flames has been pointed out previously by Dixon-Lewis and his associate^^^,^^,^^ and by Friswell and Sutton.21 The demonstration of the importance of hydroperoxyl given here has been conservative. In reality, the rate constant for reaction R5 with water as the third body is probably about a factor of 20 larger than the one used in our calculations.20 When this is taken into account the hydroperoxyl reactions become even more dominant. The chemical behavior of these flames, as noted above, bears a very strong resemblance to that of the recombination zones of premixed flames. This resemblance suggests investigation of whether or not the partial equilibrium approximation for H, 0, and OH, found to be applicable in premixed flames, might also be applicable here. To this end we have calculated values of XO(~), XOH(~), and x$) from the theoretically predicted values of xH2, xoZ,and XH~O,assuming equilibration of reactions R2, R3, and R4. These partial equilibrium mole fractions are given by x0(P)
x
= K c 2K c3 H,XO,IXH,O
X d P ) = ~ ~ c , X ~ / x ~ o o ) ~ ~ ~ ~1 1C2 ~ X H Z X(18) 0 2 1 XOH(P)
= [~c,Kc,Xs,Xo,
11 I 2
These also are plotted as a function of radial position in Figures 8-10. As can be seen from the plots, this approximation is an excellent one in the reaction zone and reinforces our analogy with the recombination zones of premixed flames. Outside this zone, however, where chemical activity is frozen and reactions R2, R3, and R4 do not have time to equilibrate, the approximation breaks down badly. This breakdown of the partial equilibrium approximation is directly attributable to the initial conditions at the nozzle lip. The persistence of hydrogen in the lean wing and oxygen in the rich central core causes eq 18 to give values for xo(P),x#), and XOH(P)which are much larger than the corresponding model predictions. We believe, however, that this is qualitatively a real effect, since it is physically necessary for a mixing zone to exist in the vicinity of the nozzle lip prior to ignition. The validity of the partial equilibrium approximation in the reaction zone extends well past the flame tip. Similar conclusions about partial equilibrium have been reached by Mitchell, Clomburg, and S a r ~ f i m in ~an~ experimental study of methane-air diffusion flames. They have, in fact, also observed significant amounts of molecular oxygen in the central core of their methane jet. In Figures 12-14 radial profiles of v&/p, the chemical production rates, are displayed for the major species H2, 02,and H20. From these and the previous discussion, a
2541
Hydrogen-Air Laminar Jet Diffusion Flames
0 0016 EQUILIBRIUM
00008
060
01
02
03
05
04
06
07
08
09
I
10
11
‘0
01
02
-I
03 0 4
RADIUS ICMI
Figure 12. Radial profile of v,&/p for reactions important in consuming and destroying H2. In the rich part of the jet, recombination of superequilibrium H atoms actually results in a net positive formation rate of the fuel. In the reaction zone, H2 is consumed mostly by H2 OH P H20 4- H, which is also the major hydrogen atom producer.
+
2
05 06 07 RADIUS iCMi
08
09
10
Figure 15. NO mole fraction compared to equilibrium at x = 1.07 cm and x = 5.10 cm. The second location is about two flame lengths from the nozzle exit. NO remains an order of magnitude below its equillbrium values well past the flame tip.
IR13) SUM12
(R14) N t 0 2 =
NOtO
(R15) O H t N S N O t H
2
‘V
0101
2~
/
151
(R31 H - 0 2 =
0 300
01
02
03
05
04
OH+O
-M Z
OH t H 0 2
(R6)
IR81 0 + H02
1
07
06
OB
HO2 + N
Z H20 t 0 2
= O2 - OH 09
10
o,,
0
Figure 13. Radial profile of vi0,bilp for the most important oxygenconsuming reactions. Where 0 and OH are most abundant, reaction H O2 M $ H02 M (R5) is the principal oxygen-consuming reaction. Where 0 and OH drop off toward equilibrium, reaction H O2 s OH 0 (R3) comes into play ( x = 1.07 cm).
+
+
+
06
-
!R21 O H + H z Z H 2 0 + H
05
IRZ1 04
i?
I\
iR61 OH + HO2
= H20
t
02-
03
2
g
02
a-
01
0
01
02
03
04
05
06
07
08
09
10
11
RADIUS ICMI
Figure 14. Radial profile of vIHIO8,/p for the most important waterforming reactions. Reactions OH H2 P H20 H (R2) and H OH -t M s H20 M (R12) are dominant in producing water ( x = 1.07 cm).
+
+
+
I 01
02
03
04
05
06
0.7
08
09
10
11
RADIUS ICMI
1 1
RADIUS 1CMI
+ +
*
(R15)
l R 5 i H * 02
-
1.1
+
consistent picture of the kinetic processes in the flame can be developed. Chemical activity is initiated by transport (both diffusive and convective) of oxygen and hydrogen into the high-temperature reaction zone separating the hydrogen jet and the ambient air. Hydrogen is consumed primarily by hydroxyl attack, forming water and hydrogen atoms. Most of the oxygen goes through an HOz intermediate, HOz being formed by reaction R5 and destroyed by free radical (H, 0, OH) attack. The primary beneficiaries of this reaction sequence are water and hydroxyl. There is also some regeneration of oxygen by this process, and a t points in the reaction zone where OH and 0 are closer to equilibrium, reaction R3 is able to compete with reaction R5 in consuming oxygen. However, because OH and 0 have their highest concentrations on the lean side of the flame, reaction R5 is the most important oxygenconsuming reaction. Replenishment of hydrogen atoms, which intiate the reaction with oxygen, is accomplished directly by reaction R2. Replenishment of hydroxyl comes through the HOz mechanism and through the reverse of
Figure 16. Radial profile of viNObIIpfor the three reactions involved in producing nitric oxide. The curve for the total NO production rate is exactly twice that for NO production from (R13). A nitrogen atom formed as product from (R13) reacts immediately with either O2 (R14) or OH (R15) to form another NO molecule.
reaction R9. Water is formed both in the initiation sequences described, through reactions R2 and R6, and in the three-body radical recombination reaction R12. Both the initiation process and reaction R12 are exothermic and deposit energy into the fluid. This is substantially augmented by hydrogen atom recombination occurring on the rich side of the reaction zone. A wide energy release zone results, significantly from the mobility of hydrogen atoms. Of some passing interest is the nitric oxide production in the flame. Figure 15 shows the NO mole fraction compared to its local equilibrium value at two axial locations, one of which is beyond the flame tip. Typical peak NO mole fractions are on the order of 50 ppm. Equilibrium is approached very slowly because of the large activation energy of reaction R13; even in the postflame-tip region, peak NO is about an order of magnitude below its equilibrium value. Equilibrium for NO is approached indirectly, the temperature dropping and reducing the local equilibrium NO concentrations far downstream. Peak NO formation rates, shown in Figure 16, occur typically at radial positions on the lean side of the reaction zone between the temperature peak and the 0 atom concentration peak, as might be expected from reaction R13. It should be noted also that the NO concentration profile ( x = 1.07 cm) is not so flat as it might appear from Figure 15. The scale for that curve is somewhat misleading, the width at half-maximum being about 2 mm. Observations and Conclusions From the results of our calculations, the following points of interest should be noted: (1) Broadening of the energy release zone and deviations from chemical equilibrium in the flame, even for the major species, result in peak temperatures which are well below the adiabatic flame temperature. The broadening of the energy release The Journal of Physical Chemistv, Vol. 8 1, No. 25, 1977
Westbrook et al.
zone is due significantly to the mobility of hydrogen atoms. Free radical (H, 0, OH) concentrations are about an order of magnitude above local equilibrium values in the reaction zone. Because of diffusion, superequilibrium is even more pronounced in the lean and rich wings. Partial equilibrium is found to be an excellent approximation for H, 0, and OH in the reaction zone; it breaks down, however, in the wings, where reaction is essentially frozen. Because of the large 0 and OH concentrations in the flame, a mechanism involving H02 as an intermediate is found to be the dominant one by which oxygen is chemically transformed in the flame.
References and Notes (1) S.P. Burke aml T. E. W. Schumann, Id. Eng. Chsm., 29?998 (1928). (2) J. A. Fay, J . Aero. Sci., 21, 681 (1954). (3) H. C. Hottel and W. R. Hawthorne, Symp. Combust. Flame Explos. Phenom., 3rd, 254 (1949). (4) A. Goldburg and S. I.Cheng, Combust. Flame, 9, 259 (1965). (5) R. E. Mitchell, Sc.D Thesis, MIT, 1975. (6) P. A. Libby and C. Economos, Int. J . Heat Mass Transfer, 6, 113 (1963). (7) R. B. Edelman et al, Symp. (Int.) Combust. [Proc.], Uth, 399 (1973). (8) F. A. Williams, Ann. Rev. Fluid Mech., 3, 171 (1971). (9) J. F. Clarke, Proc. R. Soc., Ser. A , 307, 283 (1968). (10) T. M. Liu and P. A. Libby, Combust. Sci. Techno/., 2, 131 (1970). (11) N. Peters, Int. J. Heat Mass Transfer, 19, 385 (1976). (12) R. J. Kee and J. A. Miller, AIAA Paper No. 77-639, AIAA Third Computational Fluid Dynamics Conference, Albuquerque, N. Mex.,
June 27-28, 1977 (AIAA J., in press). (13) R. A. Svehla, NASA Technical Report R-132 (1962). (14) J. H. Pohl, Masters Thesis, MIT, 1973. (15) S . Gordon and B. J. McBride, NASA SP-273 (1971). (16) C. J. Jachimowski and W. M. Houghton, Combust. Flame, 17, 25 (1971). (17) W. C. Gardiner, Jr., et al., Symp. (Int.) Combust. [ P r o c . ] , 74th, 61 (1973). (18) G. L. Schott, Combust. Flame, 21, 357 (1973). (19) D. L. Baulch, D. D. Drysdale, and D. G. Horne, Symp. (Int.) Combust. [ P r o c . ] , 7 4 7 , 107 (1973). (20) D. Gutman et al., J . Chem. Phys., 47, 4400 (1967). (21) N. J. Friswell and M. M. Sutton, Chem. Phys. Lett., 15, 108 (1972). (22) M. J. Day, K. Thompson, and G. Dixon-Lewis, Symp. (Int.) Combust., [Proc.], W h , 47 (1973). (23) G. Dixon-Lewis, J. B. Greenburg, and F. A. Goldsworthy, Symp. (Int.) Combust. [Proc.], 15th, 717 (1975). (24) W. T. Rawlins and W. C. Gardiner, Jr., J . Chem. Phys., 60, 4676 (1974). (25) A. L. Myerson and W. S.Watt, J . Chem. Phys., 49, 475 (1968). (26) W. S. Watt and A. L. Myerson, J . Chem. Phys., 51, 1638 (1969). (27) J. B. Homer and I. R. Hurle, Proc. R. Sac. London, Ser. A , 314, 585 (1970). (28) D.L. Baulch et al., Department of Physical Chemistry, Leeds, Engbnd, Reports No. 3 and 4 (1969). (29) I. M. Campbell and B. A. Thrush, Trans. Faraday Soc., 64, 1265 (1968). (30) E. Krause, New York University Report NYU-AA-66-57, June 1966. (3 1) N. N. Yanenko, "The Method of Fractional Steps", Springer-Verlag, New York, N.Y., 1971. (32) A. C. Hindmarsh, "Linear Multistep Methods for Ordinary Differential Equations: Method Formulations, Stability, and the Methods of Nordsieck and Gear", Lawrence Livermore Laboratory, UCRL-51186, Mar 20, 1972. (33) G. Dixon-Lewis, Proc. R . Soc., Ser. A , 317, 235-263 (1970). (34) R. E. Mitchell, L. A. Clomburg, and A. F. Sarofim, Combust. Flame, to be submitted. (35) Reaction numbers refer to Table I.
A Numerical Model of Chemical Kinetics of Combustion in a Turbulent Flow Reactort C. K. Westbrook," J. Creighton, C. Lund, Lawrence Livermore Laboratory, University of California, Livermore, California 94550
and F. L. Dryer Guggenheim Laboratories, Princeton University, Princeton, New Jersey 08540 (Received May 5, 1977) Publication costs assisted by Lawrence Livermore Laboratory
Calculations with a numerical model incorporatingdetailed chemical kinetics,hydrodynamic motion, and energy transport in a turbulent flow reactor have been compared with experimental results of Dryer and Glassman. A reaction mechanism, including 19 chemical species and 56 reactions, for the reaction of dilute moist carbon monoxide in air and of dilute methane in air was established for the temperature range 1000-1350 K. €402 and HzOzwere found to be important in the mechanism for both carbon monoxide and methane oxidation, and CH20,CH30, C2Hs,and C2H4were found to be important in methane oxidation. Important steps in the reaction mechanisms have been identified, and optimal values for some key reaction rates have been determined. The branching ratio between reaction 3, H + O2 = OH i- 0, and reaction 17, H + O2 + M = HOz + M, was found to be important in determining the length of the induction period in each experiment. At 1100 K the value determined for hI7was 2.6 X 1015 cm6/(mo125). Decomposition reaction 7 for HCO, HCO + M = H + CO M, was found to play a key role in methane oxidation, providing the major path for production of carbon monoxide. At 1100K, k7 was found to be 2.4 x 1O1O cm3/(mols). Even though the reaction studied was extremely oxygen rich, recombination of methyl radicals and subsequent oxidation of the ethane thus formed was found to provide a major route for methyl radical destruction. The assumption that plug flow conditions prevail in the turbulent flow reactor was examined and found to be valid under most practical conditions.
+
Introduction In systems such " engines and gas turbines, it has been recognized that quench regions may extend to temperatures Of 'This work was performed under the auspices ofthe u.s.Energy Research and Development Administration, Contract No. W-7405Eng-48. The Journal of Physical Chemistry, Voi. 8 1, No. 25, 1977
K and are responsible for major emissions of unburned and partially oxidated fuel. Furthermore, it has been noted that several key elementary oxidation reactions appear to exhibit significant non-Arrhenius behavior at the>se temperatures. Thus, for both practical and theoretical reasons, the determination of reaction mechanisms and elementary rates in the intermediate temperature range is receiving new interest.
