vvEPA
United States
Environmental Protection
Agency
Industrial Environmental Research EPA-600/7-80-018
Laboratory January 1980
Research Triangle Park NC 27711
A Theoretical Analysis
of Nitric Oxide Production
in a Methane/Air
Turbulent Diffusion Flame
Interagency
Energy/Environment
R&D Program Report
-------
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EPA-600/7-80-018
January 1980
A Theoretical Analysis of Nitric Oxide
Production in a Methane/Air
Turbulent Diffusion Flame
by
Frank E. Marble (California Institute of Technology)
and James E. Broadwell
TRW Defense and Space Systems Group
One Space Park
Redondo Beach, California 90278
Contract No. 68-02-2613
Program Element No. INE829
EPA Project Officer: W.S. Lanier
Industrial Environmental Research Laboratory
Office of Environmental Engineering and Technology
Research Triangle Park, NC 27711
Prepared for
U.S. ENVIRONMENTAL PROTECTION AGENCY
Office of Research and Development
Washington, DC 20460
-------
ABSTRACT
The coherent flame model is applied to the methane-air turbulent dif-
fusion flame with the objective of describing the production of nitric
oxide. The example of a circular jet of methane discharging into a stationary
air atmosphere is used to illustrate application of the model. In the model,
the chemical reactions take place in laminar flame elements which are
lengthened by the turbulent fluid motion and shortened when adjacent flame
segments consume intervening reactant. The rates with which methane and
air are consumed and nitric oxide generated in the strained laminar flame
are computed numerically in an independent calculation.
The model predicts nitric oxide levels of approximately 80 parts per
million at the end of the flame generated by a 30.5 cm (1 foot) diameter
3
jet of methane issuing at 3,05 x 10 cm/sec (100 ft/sec). The model also
predicts that this level varies directly with the fuel jet diameter and
inversely with the jet velocity.
A possibly important nitric oxide production mechanism, neglected in
the present analysis, can be treated in a proposed extension to the model.
111
-------
CONTENTS
Page
Abstract iii
List of Figures v
List of Tables vi
Acknowledgement vii
Sections
I Conclusions 1
II Recommendations 2
III Introduction 3
IV The Flame Structure Model 5
V The Turbulent Circular Fuel Jet 10
VI Fuel Jet with Fast Chemistry 16
VII The Strained Laminar Flame 20
VIII Nitric Oxide Production in Fuel Jet 27
IX Fuel Jet with Finite Rate Chemistry 31
X Results of Specific Jet Calculations 34
XI An Extension of the Coherent Flame Model 40
XII Concluding Remarks 44
XIII References 45
XIV Appendix A 45
XV Appendix B 51
IV
-------
LIST OF FIGURES
Figure Page
1 Elements of coherent flame model 4
2 Dependence of nitric oxide production on reactant
dilution 25
3 Fuel concentration on jet axis and integrated nitric
oxide concentration. Methane-air flame; fast kinetics
and no reactant dilution 36
4 Fuel concentration on jet axis and integrated nitric
oxide concentration. Methane-air flame; detailed
reaction kinetics and reactant dilution with reaction
products 38
5 The two-dimensional mixing layer 43
B-l Temperature, CI-U, 02, and H20 distributions in strained
laminar methane-air flame; e = 600 sec"1, K-, = K~ = 0.75
B-4 CH and H20 distributions in strained laminar methane-
air flame; e. = 600 sec"1, Ki = K2 = °'75
B-5 HO, HOz, and NOa distributions in strained laminar
methane-air flame; e = 600 sec"1, K, = K^ = 0.75
B-6 CH3 and Hz distributions in strained laminar methane-
52
B-2 COz, CO, 0, N, and NO distributions in strained laminar
methane-air flame; e = 600 sec"1, K, = <2 = 0.75 53
B-3 CHa, Na, and HNO distributions_in strained laminar
methane-air flame; e = 600 sec" , K, = <2 = °-75 54
methane-air flame; e = 600 sec"1, K, = K^ = 0.75 ,56
laminar methane-
57
air flame; e = 600 sec , K, = K? = 0.75
B-7 CHN, CH20, and NCO distributions in strained laminar
methane-air flame; e = 600 sec"1, K, = K2 = 0.75 58
B-8 H and CHO distributions in strained laminar methane-
air flame; e = 600 sec"1, K, = <2 = 0.75 59
B-9 Temperature, CH4, Oa, and H20 distributions in strained
laminar methane-air flame; e = 50 sec""1,-^ = KO = 0-75 60
B-10 0, N, and NO distributions in strained laminar methane-
air flame; e = 50 sec"1, K, = K^ = 0.75 61
B-ll COa and CO distributions in strained laminar methane-air
flame; e = 50 sec"1, K, = KO = 0.75 62
-------
LIST OF TABLES
Table
21
1 Strained Flame Methane-Air Reaction Rates
23
2 Fuel and Oxidizer Consumption Functions
3 Nitric Oxide Production Function; Dependence on ?.
Strain Rate
24
4 Nitric Oxide Production Function
VI
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ACKNOWLEDGEMENT
The authors wish to acknowledge an essential contribution of the
technical staff of the Energy and Environmental Research Corporation to
this work.
vi i
-------
I. CONCLUSIONS
A theoretical model of a turbulent diffusion flame has been developed
and applied to a methane jet burning in air. The model predicts nitric
oxide levels of approximately 80 parts per million at the end of the flame
generated by a 30.5 cm (1 foot) diameter jet of methane issuing at
3.05 x 10 cm/sec (100 ft/sec). In the model this level varies directly
with the fuel jet diameter and inversely with the jet velocity. A possibly
important nitric oxide production mechanism, neglected in the present
analysis, can be treated in a proposed extension to the model.
-------
II. RECOMMENDATIONS
Additional experimental data describing nitric oxide production in
turbulent methane-air flames of different dimensions is needed to assist
in the establishment of the scaling laws. Such data would also allow a
critical assessment of the present state of the coherent flame model and
guide its further development. It is recommended that such experiments and
the proposed extension of the model be carried out.
-------
Ill . INTRODUCTION
The coherent flame front mode"P ' is a description of fast chemical
reactions in turbulent flow in which the reactions are assumed confined to
thin flame surfaces. The turbulent flame structure then consists of a dis-
tribution of these surface elements. The model describes the manner in
which the flame elements are stretched and dispersed by the turbulent
motion, as well as the mechanism by which neighboring flame surfaces con-
sume the intervening reactant and annihilate each other. These processes
are shown schematically in Figure 1. A considerable advantage of the
coherent flame model is that it effectively separates the detailed struc-
ture of the laminar flames from the fluid mechanics so that systems with
complex chemical reactions may be treated nearly as easily as simple ones.
The work to be described here treats the methane-air flame and the
attendant production of nitric oxide. The individual flame surfaces have
internal distributions of temperature and reactants that favor the produc-
tion of nitric oxide. The nitric oxide so produced diffuses back out of
the heat zone into the cooler portions of the flame surface structure. The
production then appears as that stored in the flame surface as well as that
deposited in the products resulting from flame annihilation.
Now it is clear that the mechanism described above neglects the nitric
oxide produced "in the bulk" which, in the present picture, consists of the
products resulting from flame annihilation, slowly mixing with unconsumed
reactants. The additional production is of two parts: 1) that generated in
the products and 2) that generated in flames supported by reactants contaminated
by combustion products. In the following work, an effort has been made to
account for the second of these, that due to "re-processing" the reaction
products. The first requires a major effort to account properly for this
mechanism and, while an outline of the proposed technique is given, no
implementation of calculations have been performed.
-------
1, SURFACE STRETCHING
2, TURBULENT TRANSPORT
3, MUTUAL ANNIHILATION
Figure 1. Elements of coherent flame model
-------
IV. THE FLAME STRUCTURE MODEL
For purpose of this analysis, the turbulent flame is assumed to con-
sist of a distribution of laminar diffusion flame elements. The fuel and
oxidizer, which exist on one or the other side of the flame element, are
either the pure gases -- the injected fuel and the ambient oxidizer—
or fuel and oxidizer that has been diluted by thorough mixing with combus-
tion products.
To make this picture quantitative, we define the flame surface density
E to be the flame surface area per unit volume. Both in this concept and
in the assumptions of the flame elements, the dimensions of the flame sur-
face are very large in comparison with its thickness, so that the charac-
terization of the structure by the flame surface area is reasonable. The
local flame surface density is altered by three processes: 1) flame sur-
face production resulting from fluid straining motions in the plane of the
flame, 2) turbulent transport by which flame elements are carried from
one region of the fluid to another by large scale turbulent fluctuations,
and 3) flame annihilation, in which flame surfaces consume the intervening
fuel or oxidizer with the result that these flame surfaces vanish.
The flame surface elements that are involved in this representation
are not the usual laminar diffusion flame structures, but are dominated by
the straining motions of the reactants in the plane of the flame. In this
circumstance, the structure of the flame is no longer time-dependent, but
is fixed by the local straining rate of the fluid. Then e ^ I/time is the
p
straining rate, and D ^ (length) /time is the molecular diffusion coeffi-
cient, then the flame assumes a local thickness '\> v/D/e and consumes reac-
tants at a volumetric rate -^ /Ek per unit flame surface area. If the strain
rate e varies from point to point in the turbulent region, or varies with
time as one follows the fluid mass, then the flame will be assumed to vary
its structure in a quasi-steady manner. This theory of time-dependent
diffusion flames in a straining gas motion is given in appendix A.
-------
When the flame surface density S and the strain rate e are known
locally, the reactant consumption rates within this region are known also
because they are fixed by the flame structure. The actual flame structure
may be calculated to any degree of approximation between the simplest,
where the chemical reaction rates are very rapid so that the reactant con-
sumption is controlled by molecular diffusion, to the detailed structure
involving the complete chemistry with appropriate rates. The flame struc-
ture with finite rates requires numerical calculation but, because it is a
steady one-dimensional problem, the actual calculations may be accomplished
with relative ease.
The appropriate form of the equation describing the flame surface
density may be deduced from first principles by considering the distortion
and migration of surface elements, fixed to the fluid, in a turbulent
(2^
medium.v ' Here, it will be sufficient to motivate the form by physical
reasoning and then to suggest the manner in which the various terms scale
with features of the flowfield and the chemistry of the flame structure.
Now in a fluid with mean velocity components U^, the expression
Dt - 8t "j 3x,
J
gives the change in flame density following a mean fluid element. Accord-
ing to our model described earlier, this change may be written in the
following form
— = turbulent diffusion of flame surface into the region
L/ \f
+ increase of individual surface element area by
turbulent straining motions
- reduction in flame surface resulting from local consump-
tion of one of the reactants.
In these calculations, the turbulent diffusion of flame surface will be
described using a turbulent diffusivity and the assumption that the
-------
turbulent fluctuation velocities are large in comparison with molecular
diffusion velocities, familiar from usual turbulent treatments of heat and
mass transport. Thus, if we denote D the turbulent diffusion coefficient
that arises in the description of momentum exchange, the turbulent diffusion
term applicable to the treatment of the turbulent circular jet, which we treat
as our example, is,
-
r 9r
where the appropriate boundary layer approximation has been made. The rate
of increase of an element of flame surface area is proportional to the
strain rate in the turbulent fluctuations. Under the assumption that this
strain rate is proportional to the rate of strain in the mean motions, the
rate of increase in flame surface density is thus proportional to the
product of the strain rate of mean motion and the local flame density. Then,
calling a the unknown constant of proportionality, we take the second term on
the right equal to
e E = a
3r
where, again, we have written the term in the form appropriate to the cir-
cular jet, W being the velocity component of the mean motion in the
direction of the symmetry axis.
The general nature of the process by which flame surface is removed
from the field is best pictured by considering two neighboring laminar
diffusion flame fronts parallel to each other and containing one constitu-
ent, say, fuel, between them. As the motion progresses, the intervening
fuel is consumed and both elements of flame surface are extinguished. A
similar process takes place if the intervening constituent is the oxidizer.
To make the mechanism quantitative consider a volume containing many flame
elements. The fraction of local volume occupied by fuel is proportional, under
our assumption of constant density, to the mass fraction K, of fuel. Moreover,
the rates of reactant consumption by an element of laminar flame are presumed
known from detailed calculations of that flame structure. If we call v, the
volume rate of consumption of fuel per unit flame area, the rate at which
volume occupied by fuel is being consumed in a unit volume of space is v, E .
-------
But that unit volume of space contains only a fraction K, of fuel so that the
rate of fuel consumption in a unit volume of space divided by the amount of
fuel contained in that volume is v. E/K, . Thus, if the flame surface area is
nearly uniformly distributed over the region, this expression gives also the
fractional rate of flame surface annihilation so that the flame surface area
reduction rate due to fuel consumption is
In a completely similar manner, we may reason that the flame surface area
reduction due to consumption of oxidizer is proportional to
v
— £
and because in a given region, individual surface elements are being
removed by either the exhaustion of fuel or of oxidizer, the two expressions
will be considered independent and additive.
If now we collect the various terms that have been discussed, it is
possible to write in detail the conservation equations for flame surface
density
The form of equation has been chosen to be that appropriate for the analysis
of the circular jet.
The conservation equations for the individual species are conven-
tional except for the terms describing the reactant consumption by chemical
reaction. These will be given in terms of the flame surface density and
the reactant consumption for a unit flame surface supplied by one-dimensional
flame calculations. For the fast chemical reaction, we need consider only
three constituents, the fuel KI , the oxidizer <2, and the product K_,
related by the fact that the sum of mass fractions is unity
-------
so that only two of these, K, and K^, need be treated in detail. The
actual chemistry, in contrast to the overall reaction between fuel and
oxidizer, is contained within the one-dimensional flame structure.
The consumption rate of fuel per unit volume is simply the product of
the effective influx velocity v,, the flame surface density £, and the mass
fraction K,* of fuel in the fuel-containing constituent of the turbulent
structure. The conservation equation for fuel may be written down directly
as
Similarly, the conservation equation for the oxidizer component is
• K * v0 I (4)
It remains to define the consumption rates v, and v,, for the fuel and
oxidizer components. Regardless of whether these quantities are defined
through use of infinitely fast reaction rates or by detailed calculation
of the one-dimensional flame structure, they depend upon the reactant
concentrations on each side of the flame, K, and K^, and the local strain
rate e of the mean flow. It is this latter item which couples the local
diffusion flame structure to the gasdynamic structure.
-------
V. THE TURBULENT CIRCULAR FUEL JET
The detailed solution for a turbulent flame structure described in the
manner outlined in the previous section requires a turbulence model for the
mean flow and, because there is a choice of turbulence model to be used, it
is preferable to go directly to the example of interest rather than describe
the procedure in general. For the problem of the circular fuel jet, we
shall choose the elementary model utilizing a scalar turbulent diffusivity.
Furthermore, we shall, in the interests of simplicity, neglect the change
of mean gas density associated with the chemical reactions. This restric-
tion in no way implies that the density change is a negligible factor.
Rather the use of a turbulence model for flows of non-uniform density intro-
duces a degree of uncertainty of its own which makes it additionally
difficult to judge the merits of the flame model.
Under these restrictions, the gasdynamic field is described by the
equations
-------
2 2
flow, has the dimension of (velocity) x (length) . And because D has the
dimension of (length) x (velocity), there being no other characteristic
dimensions in the problem, it is evident that the turbulent diffusivity is
proportional to the square root of the jet momentum, and hence is a constant
It is possible to find a similarity solution for the turbulent jet, or
more properly for a point momentum source. To carry this out, it proves
convenient to introduce a modified radial dimension
n-Vf^ <8>
where u is the constant jet momentum flux
= f W2 rdr (9)
u
"0
The stream function ty, with the usual properties that U = - — -^- ,
T r\ i I O Z
W = —•-£•, may then be written in the form
i O i
ty = Dz F(n) (10)
Substitution into the momentum equation, Equation 6, yields the ordinary
differential equation for F(n)
-{- FF" + - F'F1 - -1* FF'>'= ^- (F" - 1 F1
In n 2 j dn \ n
which may be integrated once to give
.IFF- = F" -Ip1 (n)
the constant of integration being zero because — F1 and F" vanish for
large n- Fortunately, the solution for Equation 11 may be written simply
as
11
-------
1
4
(12)
which satisfies the requirement that the stream function vanish on the
symmetry axis, n = 0, and satisfies identically the momentum conservation
r 2
relation y = J W rdr, as
0
n dn = 1 (13)
The velocity components of the mean flowfield are then
= 3 U.
and
l 2,2
U = - !/£- u - - -
1 2,
Now explicit solution of the flame density and reactant conservation
relations require a quantitative form for the reactant consumption rates.
If the chemistry were actually of infinitely fast rate and if the mutual
reactants were uncontaminated by products of combustion, the values of K^
and K*V£ appearing in Equations 3 and 4 could be written
where D is the coefficient of molecular diffusion, e =
a
is proportional
to the turbule'nt strain rate, and 3-|, 62 are constants for a given chemical
12
-------
combination, depending upon the reaction stoichiometry. Numerical calcula-
tions of strained laminar diffusion flames, which will be discussed later,
suggest that for a wide range of flame straining rates and reactant dilu-
tions, the values of K-|V-| and K^V,, may be written
KI*VI = B1 G-1(K1,K2) /DF ; K2*v2 = B2 G2dc.,,ic2) v^te (17)
where G-,(K-j,i<2), G2(<, ,<„) depend only upon the local average reactant mass
fractions and G-j = G2 = 1 for fast chemistry. The equations for reactant
consumption and flame density may then be written
w
u L = J- ° (v> n -1 R R ^ ^ ^ /pf) r MQ^
^T V» 3 V I ^ 1 V 1 PoaO\K-1»K-O/C"L''< V1-7/
(20)
(21)
When the chemistry is fast, so that G, = G2 = 1, it is convenient to
use a linear combination of K-, and K- that satisfies a homogeneous diffu-
1
sion equation. Such a combination is <-, + , . 0 ,0 KO = J, and because
I I T PO/ PI O
a diffusion flame with fast chemistry utilizes fuel and oxidizer in the
stoichiometric ratio <$>, the quantity 3o/3-i = <|) and
W "~" K- T Tii l^> *•
^3
Thus J(r,z) satisfies
13
-------
with the conditions J(-iZ) = o and hence, comparing Equations 23, 6, and 7,
it is clear that J(r,z) -v, w(r.z). The constant of proportionality between
J and W is easily found by noting that if the actual flow of fuel injected
into the stream is y, then the flow of the product formed is (1 + *)Y for
large values of z where the reaction is complete and < = 0. Then the
integrated flux of the product is, as z
->- 00
/ Kr> W rdr = y(l + $) (24)
Jo
00
and because integration of Equation 23 shows that y* J W rdr is conserved
along the z direction, it follows that
oo
f J W rdr = y (25)
Because J is proportional to W, the integral is the same as that evaluated
to ensure conservation of momentum with z and hence it follows, comparing
Equations 9 and 25, that
(26)
We have then the algebraic relation
<27>
in addition to the relation Equation 21 and, as a consequence, we need only
work with a single species conservation Equation 19.
In order to make this identification between J(r,z) and W(r,z), it has
been necessary to introduce the volume flow rate y as well as the momentum
y of the jet. These two quantities may be used to characterize the jet
velocity
velocity ^ y/y (28)
14
-------
and the characteristic dimension of the jet,
length ^ Y//M (29)
when, in fact, the length Y//U = /if R, where R is the initial radius of a
jet with uniform velocity.
15
-------
VI. FUEL JET WITH FAST CHEMISTRY
When the chemical reaction rates are fast, the relevant equations,
Equations 18, 27, and 20 are simplified principally by the fact that the
functions G-,, G2 are both unity and, in general, &2 = B-,. Now if there
were no reactions and if the flame straining were neglibible (e = 0),
then the problem would possess similarity solutions of the form K, ^ — k,(n)
and E '\j — a(ri). There is a certain advantage in writing these quantities
in that form, as
and
= ill
l 8 Dz
3,2
a(n,z)
(30)
(31)
Now because we shall use an integral technique to obtain our final
result, it is convenient to rewrite Equations 18 and 20 in similarity
variables and integrate them over the flame cross section. Introducing
the similarity coordinates, n and z, we have
-- ^ U
4r (r W
- (n r W K ) + W
and
D z
/3~
a
8r
1 IT
r E - A
r Z
9W.
3r
r Z
K-.K,.
(32)
(33)
16
-------
Integrate these from n = 0 to n = °° and take account of the behavior of
K and E for large radii. We then have from Equation 32
7TF / IV) k] (n,?) n dn = - (|) / ^ (- £-) a(n,O n dn (34)
0 V '
where we have introduced a new variable for distance along the axis,
S=f°z (35)
Treating the flame density relation in the same manner
o n(C,n) dn - - - ,(n.e)n
0 0
where in this notation
i I r I \
(37)
To complete the solution using the integral technique, it is required
to select profile shapes for the mass fraction distribution and for the
flame surface density. These are chosen specifically in the following
form:
(38)
))
f(?) (39)
17
-------
- g-) hU) (l - f(c)) [l - (EL) 1, (,
(40)
where f(?) and h(£) are unknown functions of distance along the axis and
will be determined through the integral relations, Equations 34 and 36.
Without justifying this choice in detail, several properties are important:
a) They are exact when there is no chemical reaction.
b) They satisfy exactly the conditions given by Equation 27
among species mass fractions.
c) They satisfy the appropriate boundary conditions at the
symmetry axis and at distances remote from the axis.
d) The flame surface density vanishes, as it must physically,
where either the fuel or oxidizer reactants vanish.
Now the value of F(n) = n2/0 + ^ n ) » Equation 12, is the exact solution
for the stream function in the jet problem utilizing the turbulent diffu-
sivity formulation. Utilizing this and the representations for k^n,^),
Kol1"!^)' a(n,O given in Equations 38, 39, and 40, the integrations indi-
cated in the integral relations, Equations 34 and 36, may be carried out.
The integrals are of three classes: (i) those which may be carried out
in an elementary manner, (ii) those which may be reduced to an integral of
(1+1 2)~n and have the value
i
and (iii) those of the form
~/F'\n . l K
(42)
These may, after a transformation, be written as
18
-------
In= (2)n+3B(2n + 2 +1 ; 5/4) (43)
r(2n + 2 + {)r(5/4)
n+3 -±—j ^^ (44)
, + 3 + £
where both the Beta function and Gamma function representations are
tabulated.
If we carry out this somewhat lengthy task on Equations 34 and 36, a
pair of non-linear ordinary differential equations results
^P (1-f) = -{0.44424 - 0.41251 ( ' V ) ? 0-f)h (45)
d_, _ f (l-f)h = 0.45090 - 0.47344
- X (l-f)h2 (0.25386) l - L (46)
These are to be integrated from £ = 2, the virtual origin of the jet being
at C = 0, to an arbitrary value of £; the initial values of f(£_) = 0 and
h(f ) = h , where h gives a measure of the amount of flame surface that is
v^o o o
present at the start of calculation. Physically, the initial value of
is related to an ignition process which the turbulent flame model does not,
of course, describe. It is to be expected that varying the initial value
h will have a significant effect upon the local structure of the flame,
but a much smaller effect upon the development far from the origin. The
values of and -£- are fixed by the reactants and by the turbulence model,
respectively. The values of a and X, on the other hand, are numerical
constants which are universal in the sense that they do not depend upon
stoichiometry, momentum flux, or geometrical size of the problem.
19
-------
VII. THE STRAINED LAMINAR FLAME
This section describes the strained laminar flame calculations which
provide the reactant consumption and nitric oxide production rates used in
the model. When the chemical rates are taken to be infinitely fast, an
analytical solution yields the fuel and oxidizer rates written in Equation
16 and applied in Section IV:
K* v1 = 31 vW ; K2 V2 = B2 ^ ^47^
In the more general case in which fluid properties are variable, heats
of reaction and kinetic rates are finite, and many chemical reactions are
treated, a numerical calculation is required. The results presented in this
section come from such a calculation and are the work of T. J. Tyson and
his associates at the Energy and Environmental Research Corporation. The
authors acknowledge with thanks their cooperation in making the results
available and thank especially C. J. Kau who made, under difficult circum-
stances, the many computer runs we requested. The set of kinetic equations
describing the methane-air reactions in the computer program were developed
by the Energy and Environmental Research Corporation staff under Environ-
mental Protection Agency Contract Number 68-02-2631 and were reported in
their Monthly Progress Report No. 14, June 1978.
For the reasons given in Section III, solutions are needed for a range
of strain rates and for various dilutions of the fuel and oxidizer by com-
bustion products. The thermodynamic states of the diluted reactants are
those reached when products in the form of C02, H20, and N2 are added
adiabatically to the reactants and no chemical reaction is allowed.
The combination of strain rates and dilution ratios for which calcula-
tions were made and the results are given in Table 1. (Several cases were
omitted initially due to the lack of time, but were determined to be not
needed when the results were examined.)
20
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Table 1. STRAINED FLAME METHANE-AIR REACTION RATES
CASE
1
2
3
1
% AIR
1.0
1 0
T 0
4 1.0
6
7
8
10
11
12
14
15
16
0.75
0.75
0.75
0.50
0.50
0.50
0.25
0.25
0.25
% FUEL
1.0
0 75
0 50
0.25
0.75
0.50
0.25
0.75
0.50
0.25
0.75
0.50
0.25
STRAIN RATE,
SEC"1
50
250
500
50
REACTION RATE, MOLES/CM2-SEC
CH4
2.9 • 10"5
6.68
9.22
2.32
200 4.60
50 1.96
200 1 3.98
400 6.0
600 7.43
50 1.57
°2
4.75 • 10"5
10.4
13.6
3.89
7.56
3.3
6.45
9.07
10.9
2.65
200 ' 3.24 i 5.23
400 ; 4.74 7.32
600
50
200
400
600
50
200
400
600
50
200
400
600
50
200
600
50
200
400
600
50
200
400
600
50
200
400
600
50
200
400
600
50
200
400
600
5.89 8.90
2.02
4.78
6.22
7.69
1.71
3.60
5.26
6.53
1.38
2.87
4.18
5.20
1.59
3.39
6.13
1.35
2.86
4.20
5.22
7.11
2.31
3.38
4.21
1.02
2.21
3.25
3.99
0.88
1.88
2.77
3.44
0.75
1.55
2.27
2.83
3.27
6.44
8.96
10.8
2.80
5.51
7.75
9.33
2.26
4.46
6.25
7.58
2.44
4.79
8.0
2.09
4.13
5.78
6.46
1.72
3.41
4.77
5.76
1.4
2.72
3.77
4.45
1.21
2.37
3.30
3.94
1.02
2.01
2.80
2.35
NO
3.5 • 10"9
1.04
0.42
4.19
1.56
5.99
2.47
1.28
0.94
9.3
3.98
2.63
1.91
5.57
2.41
1.25
0.77
7.34
3.45
2.01
1.35
10.5
5.17
3.21
BT a.
0.142
0.146
0.143
0.114
0.113
0.096
0.097
0.104
0.105
0.077
0.079
0.082
0.083
0.099
0.105
0.108
0.109
0.084
0.088
0.091
0.092
0.068
0.070
0.072
2.40 0.073
7.14 '•; 0.078
3.66 : 0.083
1.68 ; 0.087
8.69 : 0.066
4.69 : 0.070
3.10 0.073
2.31 0.074
11.1
6.24
4.31
3.33
8.46
5.19
3.77
2.94
9.52
5.95
4.43
3.58
11.2
6.93
5.25
4.35
0.054
0.057
0.058
0.059
0.050
0.054
0.056
1&Z G2
0.232
0.228
0.210
0.19
0.185
0.161
0.158
0.157
0.154
0.130
0.128
0.127
0.126
0.160
0.158
0.155
0.153
0.137
0.135
0.134
0.132
o.m
0.109
0.108
0.107
0.119
0.117
0.113
0.102
0.101
0.100
0.098
0.084
0.083
0.083
0.081
0.069
0.067
0.065
0.056 \ 0.063
— . . 4
0.043 • 0.059
0.046 ,: 0.058
0.048 ; 0.057
0.049 I 0.054
0.037
0.038
0.039
0.040
0.050
0.049
0.048
0.047
21
-------
In Section III, Equation 17, the fuel and oxiclizer consumption rates
were written in the form:
1 -| 1-,,2
The functions 3] GI and 32 &2 are a1so given 1n Table ] and' for each C0m"
position, are indeed almost independent of the strain as they are, exactly,
when the reaction rates are infinite. The average values of these functions
for each dilution ratio are collected in Table 2. Only the function for
the fuel is needed and is represented with acceptable accuracy by the
expressions:
= O.H2 (49)
-,
(50)
-1/2
empirical. It was noted that this rate decreased approximately with e ' .
The treatment of the nitric oxide production rate is somewhat more
rical. It was noted that this rate decreased ap
We introduce, therefore, the variable H/T defined by
1/2
p /D
(51)
where (LNO is NO production in mass per unit flame area per unit time, and
observe that the dependence of the NO production on strain is represented
within about a factor of two in this way. The values of H/T for cases 3
and 12, shown in Table 3, are representative examples. Average values for
all compositions are given in Table 4 and were found to depend, approxi-
mately, linearly upon the variable (KI + K2), as Figure 1 shows. The fit
to this data used in the analysis in Section VI is
1= 4 x 10"3 sec"1 (52)
H = {2 - (K + K)> (53)
22
-------
Table 2. FUEL AND OXIDIZER CONSUMPTION FUNCTIONS
&i G,
I I
~>\
1.00
0.75
0.50
0.25
1
0.142
0.123
0.098
0.060
0.75
0.113
0.105
0.084
0.053
0.50
0.100
0.088
0.046
0.046
0.25
0.080
0.070
0.038
0.038
32G2
^\
1.00
0.75
0.50
0.25
1
0.44
0.376
0.246
0.150
0.75
0.38
0.314
0.232
0.132
0.50
0.31
0.268
0.20
0.114
0.25
0.256
0.218
0.166
0.096
23
-------
Table 3. NITRIC OXIDE PRODUCTION FUNCTION;
DEPENDENCE ON STRAIN RATE
Case
12
e
50
200
400
600
50
200
400
600
3
H/T x 10°
27.5
22.7
16.6
14.9
50.9
57.3
55.9
52.9
Table 4. NITRIC OXIDE PRODUCTION FUNCTION.
Case
1
2
3
4
6
7
8
H/T x 103
1.0
1.4
2.0
3.5
2.0
2.9
4.5
Case
10
11
12
14
15
16
H/T x 103
3.0
4.2
5.6
4.8
5.6
6.6
24
-------
o
I I
0.2
0.4
0.6
0.8
.0
Figure 2. Dependence of nitric oxide production on reactant dilution
It is significant that the chemical kinetics code from which these
calculations were made, gives the result, shown in Figure 2, that NO
production rate actually increases as the reactants are diluted with hot
combustion products. This result might not have been expected, but because
examination of this code was not the aim of the present work, this interesting
result was not examined in detail. One may speculate that the increase of
flame temperature accompanying reactant dilution and the "re-processing"
of combustion product in the hot flame contribute to this trend.
The computations which yield the above discussed rates also determine
the detailed structure of the laminar flame. A complete set of concentration
profiles and the temperature distribution for case 6 are shown in Figures B-l
through B-8 in Appendix B. Figures B-9 through B-ll contain a few results
for the same reactant compositions, but a lower strain rate. Since a
25
-------
thorough examination of these results remains to be done, no further
analysis will be presented at this time.
26
-------
VIII. NITRIC OXIDE PRODUCTION IN FUEL JET
If we denote by KN the mass fraction of nitric oxide, we may write the
nitric oxide conservation (or production) relation as
(54)
where the rate relation for the production of nitric oxide, developed in
the preceding section, has been employed. In this expression, I/T is an
absolute rate constant, and H(K, ,Kp) is the function that accounts
for the effects of free stream reactant dilution upon nitric oxide
production within the flame. If Equation 54 is rewritten in the form
|f (r U K||) * |j (r H *„) - fp (r D) «. 1 H^) ^ r t (55)
use of the simTlarity variables and the stream function leads to the result
(56)
where we define a = 1/r/a.
00
Now we wish to calculate the ratio of nitric oxide flux f p W KM(2Trr)dr
» 0 N
to the total product flux J p W (27rr)dr and this ratio becomes
0
27
-------
OO '-*-' ^^
/ p W KN (2frr) dr // p W (2irr) dr = / KN (^
0 7 0 0 V
dn
\ dn
But the integral in the denominator is equal to 4/3 and, hence,
$N = -
0
(57)
Thus if, returning to Equation 56, we integrate this over the jet cross
section and take account of behavior of the stream function at large radii
or
ld_
Idz
• (f'
£z2D2
dl zTN
!./ L.\
^- n;
-1/2
H s
?
d / F'
1/2
H E n dn (58)
Now this expression may be put in a more suitable form for integration by
introducing the dimensionless distance along the axis, £, Equation 35, and
the representation of the flame surface area, Z, given by Equation 31.
The resulting relation
-1/2
n dn
(59)
contains a dimensionless group (2/3^ (y^) which will be of considerable
physical significance in developing the scaling laws for the production of
nitric oxide in flame structures. The volume flow, y» and momentum flow, y,
28
-------
may be represented in the forms
where WQ is the effective average velocity from a jet of radius R. Sub-
stitution of these expressions into Equation 59 yields the equation for
the nitric oxide concentration
where d = 2R is the initial jet diameter.
The representation of a(n,5)> appropriate to the integral solution,
is given by Equation 40, and the function H(K, ,Kp), developed in the pre-
ceding section, will be written
H(
-------
In a manner similar to the procedure used in evaluating certain of the
integrals in the flame surface density earlier, each of these integrals
is of the form
•,\H/2/F,\n
_U /£_) n dn = Jn (63)
I ' ' \ ' / I \ /
0
where we know F(n) so that
-2
With some variable transformation and rearrangement, we may find that
( '
and, thus, the integrals may be evaluated numerically with the aid of
tabulated values of the real gamma function.
If these numerical evaluations are carried out and we define a variable
PN, proportional to the nitric oxide formation,
(65)
The differential equation for P.. is then
d P..
!1 _ _L D
i >. >- r
N {°-058308 5 + 0.063842 $ (1 - f)
- 0.091893 (1 + 4,) + f) (1 - f)h (66)
which is to be solved in conjunction with Equations 45 and 46 for the
reacting jet structure.
30
-------
IX . FUEL JET WITH FINITE RATE CHEMISTRY
The calculations for the detailed structure of methane flames, dis-
cussed in Section V, permit a somewhat more realistic description of the
turbulent flame structure than that afforded by the fast chemistry employed
in Section IV. In addition, because these calculations were carried out
with various degrees of reactant dilution with combustion products, it is
possible to account for the dilution of the unburned reactants which
takes place as one proceeds along the jet. It will be recalled that the
model described in Section IV, the combustion rate was entirely diffusion
controlled and the reactant concentration far from the flame structure is
assumed to be that of the uncontaminated reactants supplied to the jet and
the oxidizer field. This is certainly a limit, in one sense, because the
consumption per unit area of the flame is a maximum and, hence, the total
flame surface area required to consume the fuel in the jet, is a minimum.
It is reasonably evident, then, that the nitric oxide produced within the
flame itself is well below the actual physical value, if not a minimum.
Another sort of limit that may now be considered follows from the
assumption that the combustion products, which are produced in burned out
portions of flame, are immediately mixed with the reactants. In this limit,
the values of K,, K~ with which to evaluate the laminar flame structure
are the local mean values which we employ in our flame model.
The fact that extensive numerical calculations of Section V show that
the reactant consumption rate per unit area is proportional to yfk permits
the representation for fast chemistry to be carried over not only for finite
reaction rates but also for contaminated reactants. Referring again to
Equation 17, we select
31 = 0.142 (67)
GI(KI,KZ) = y K-, + y KZ (68)
31
-------
on the basis of the calculations described in Section V. Thus, omitting
the details of a calculation which parallels quite closely that of Section
IV, the integrated fuel consumption and flame surface density equations
are
dn (69)
\0/ J iun \ U /I \/ i / <-/
0 v ;
and
1 I ~ 1 f > " I • 1\ " V I »^/ I ±1 ,, ± "
~Al77| f \ j._ I ~ _ I? i, .. I i KI ~ -,
which are the counterparts of Equations 34 and 36 that were valid for
undiluted reactants. The forms of k, , K^> °> given by Equations 38, 39,
and 40, and the known function F(n) allow the explicit evaluation of the
integrals that occur utilizing again the results given by Equations 42
through 44 which were useful in developing the equations for the fuel jet
and fast chemistry. These relations are then, in detail required for
numerical computation,
(71)
where a new variable
±A^t)](1.f)h (73)
has been introduced. The variable coefficients are
32
-------
11 1 + J> f / l + i 1 +
A = 0.25386 - H - £— + o.17679 (Jzf 41 + <(» f - 4 E (74)
B = 0.45090 (l - |i-!_LA-L) (75)
(76)
It will be found, upon detailed inspection, that Equations 71 and 72 reduce
to Equations 45 and 46, respectively, when the second term in A, Equation
74, is deleted.
33
-------
X . RESULTS OF SPECIFIC JET CALCULATIONS
The model of turbulent chemical reactions in fuel jets, which has
been presented, rests upon three empirical constants which, if the physics
of the model were accurate, would be universal constants. While this
universality is not likely, it is to be hoped that the results will be
general for turbulent fuel jets and for flows that are reasonably similar
to fuel jets. The first of these, which in our representation appears as
/M/D , is well known and valid for all turbulent jets and, as a matter
of fact, does not enter explicitly in our calculations. The second empir-
ical constant is a /JI/D which enters in connection with the turbulent
straining of the flame surface area. It is well known that the turbulent
straining rate in a field of inhomogeneous turbulence is related to the
mean flow in a manner that depends upon the type of field (i.e., turbulent
jet, mixing region, boundary layer, etc.) but is the same for similar flow
fields. Thus, when we assume the turbulent straining of the flame surface
to be directly proportional to the straining rate of the mean flowfield,
we acknowledge an unknown constant of proportionality, but one that will
be of the same value for all turbulent fuel jets. This proportionality
appears in the form a vy/D.
The flame shortening process is probably the most controversial of the
features in the turbulent combustion model which we have presented. If the
physical picture, based upon the consumption of regions of fuel or oxidizer,
is substantially correct, then there is a constant of proportionality which,
for a specific type of turbulent flowfield, connects the actual annihilation
of flame surface to the related dimensionless quantities which result from
our model. This constant appears as X in our calculations.
Now even a cursory appraisal of the equation describing the development
of the flame surface area suggests that the values of a /jI/D and X are
intimately related to the beginning and end of the jet, respectively.
When the flame is first ignited, for low values of £, the value of a v^/D
determines both the rate of growth of flame surface area and the reactant
34
-------
consumption per unit area. But as the flame density grows and regions
within the flame develop where one of the reactant concentration values is
low, the terms describing the annihilation of flame surface become impor-
tant and balance, to some extent, the flame surface growth resulting from
turbulent straining. A sort of balance is reached between the straining
and annihilation terms, respectively, proportional to a v'y/D and to A,
which sets the rate at which reactants are consumed in the jet. Together
they determine the length of the flame required to consume the fuel. The
two quantities also determine the growth of combustion intensity at the
beginning of the flame and the point along the flame length at which the
combustion intensity reaches its peak.
Because these quantities are readily observable from experiments with
gaseous fuel jets, the suitable values for the parameter have been deter-
mined in previous work to be approximately
X = 5.0
= 2.0
These values are based largely upon the extensive experiments of Hawthorne,
( 1)
et al.v ' These same data have allowed a satisfactory check of the vari-
ations of flame length with fuel stoichiometry.
Using the model described, together with values of the two constants
thus determined, calculations have been made for the production of nitric
oxide in fuel jets burning methane in air at atmospheric pressure and
temperature. The first of these calculations used Equations 45 and 46,
describing the fuel jet with fast chemistry and undiluted reactants,
together with the differential equation, Equation 66, describing the pro-
duction of nitric oxide within the flame structure. The interesting result
of this computation is shown in Figure 3 where quantities proportional to
the center! ine fuel concentration and the integrated average nitric oxide
35
-------
2.0
00
0>
TD
Ol
03
O)
S-
I
c
O
c:
4-
O
u
fO
S-
X
(O
O
O
4->
IO
O)
u
O
O
Figure 3. Fuel concentration on jet axis and integrated nitric oxide concentration.
Methane-air flame; fast kinetics and no reactant dilution.
-------
concentration are shown as functions of the distance from the virtual
origin of the jet measured in terms of the initial jet diameter. The
quantity 1-f, referring to Equations 30 and 38, measures the centerline
fuel concentration as a fraction of the value that would be observed if
there were no chemical consumption of fuel, but only that dilution which
would result from mixing with the ambient gas. This, if 1-f remained at
the value unity, no chemical reaction would be taking place. Conversely,
when 1-f appraoches zero, it signifies that the fuel has been consumed,
not simply mixed with a non-reacting diluent, until it is scarely detect-
able. The nitric oxide concentration P.., defined by Equation 65, increases
very rapidly at the start of flame, levels off, and then drops slightly
as the flame end is approached. To interpret this behavior physically,
one must recall that the nitric oxide concentration is averaged over the
entire jet including all entrained but unburned oxidizer. The initial
rise parallels that of the flame surface density because, until the flame
annihilation mechanism begins to be important, the nitric oxide is retained
within the flame structure. The subsequent tendency to level off results
from the gradual decrease of flame surface density as the fuel is consumed,
as well as the continued entrainment of air into the jet structure. The
final decrease in nitric oxide concentration results from the mixing of
the existing nitric oxide and additional, non-reacting oxidizer.
The maximum value of P.. attained is about unity, and the physical
value of concentration may be estimated using Equation 65. Recalling the
calculated values k = 5.66 x 10"' sec" and B-i = 0.142, we find that the
actual concentration is
*„ = 4.0 x 10"3 (£-} PM (77)
so that for a one foot diameter burner with an efflux velocity of 100 feet/
<\, -5
second, $.. = 4.0 x 10 or 40 parts per million.
The results of a corresponding calculation are shown in Figure 4 for
the model in which the laminar diffusion flame calculations are based upon
the local mean reactant concentrations rather than upon the uncontaminated
values. The results consequently account for the lower reactant consump-
tion rate per unit flame area, the greater flame surface density which
37
-------
CO
co
OJ
rs
ro
1.0
2.0
O
c;
o
o
x
ro
o
c
o
tT3
S-
+J
c:
O)
o
c:
O
o
0.8 -
0.6 -
0-f)
0.4 -
0.2 -
40
Figure 4. Fuel concentration on jet axis and integrated nitric oxide concentration.
Methane-air flame; detailed reaction kinetics and reaction dilution with
reaction products.
-------
develops as a consequence and the greater production of nitric oxide.
This system is defined by the differential equations, Equations 71 and 72,
and the more complicated coefficients given by Equations 74, 75, and 76.
Again, the production of nitric oxide is described by the differential
equation, Equation 66.
The fuel concentration, as appears by comparison of the two figures,
is not strongly affected by the inclusion of reactant vitiation so far as
either the length of the flame or the concentration distribution is con-
cerned. This result illustrates very well the manner in which fuel con-
sumption rate by a flame element and the total flame surface area within
the jet envelope, compensate so that this product is essentially constant.
The nitric oxide production, however, responds nearly proportionally to
the flame surface density and, as a consequence, is higher for the flame
structure that accounts for vitiation. In this case, the nitric oxide
production is nearly doubled from jet calculations utilizing uncontamin-
ated reactants.
The scaling law for nitric oxide production is contained in
Equation 65:
N 1 0
in which 3, and £ are constants and P.. is independent of the fuel jet diam-
eter and velocity. The prediction, then, is that the nitric oxide concen-
tration at the end of the flame varies linearly with diameter and inversely
with the jet velocity.
Sufficient experimental data is not yet available for critical compar-
ison with the predicted nitric oxide concentration values or with the scal-
ing law. The observed levels, however, are considerably higher than
those given by Equation 78 and appear to have a weaker dependence on jet
diameter than this equation predicts. The difference in concentration
level may come from the reactions, neglected in the analysis, which take
place in the burned-off portions of the flame. The extension of the
model described in the next section is intended to account for these
additional reactions.
39
-------
XI. AN EXTENSION OF THE COHERENT FLAME MODEL
The coherent flame model, as originally formulated and as applied in
the preceding sections, assumes that chemical reaction rates are fast rela-
tive to mixing rates. This section outlines an extension of the model which
allows both fast and slow reactions to be treated. The motivation for such
an extension is the need for a description of the nitric oxide production
which occurs in the methane-air flame after the more rapid energy producing
reactions are complete.
In the model, as applied in Sections VI and VII, all reactions take
place in the strained laminar flame. When these are all fast, the species
consumption and production rates are equal to the rate per unit flame area
multiplied by the flame area per unit volume. That is, there are no further
reactions in the products generated when portions of the flame sheet burn
off in the flame shortening process. Reactions that are slow, however, such
as those which lead to nitric oxide, may continue in such regions. It is
the purpose of the model extension to account for these additional reactions.
It is clear that for a kinetic system of any complexity, such as that
for methane-air, the reactions taking place in the product gases must be
treated numerically as they are presently in the strained flame. Such a
requirement necessitates a simplification in some aspects of the model if
the computational effort is to remain reasonable. A somewhat different view
of the turbulent mixing process, about which there is some disagreement
(even between the present authors), appears to provide sufficient simplifi-
cation. The approach is suggested by the many experiments on mixing layers,
for example, those of Brown and Roshko' ' and Konrad^ ', and, more recently,
40
-------
on turbulent jets/6) which may be interpreted to imply that it is in the
breakdown of large-scale structures that molecular mixing and the subse-
quent chemical reactions occur, i.e., that the reactants are entrained into
the layer and jet by large inviscid motions and subsequently mix as the
large structure breaks down. If this interpretation is correct, it suggests
the approximation that mean quantities be taken as independent of the trans-
verse, y, or radial, r, direction and that attention be concentrated on the
axial (or, in a sense, temporal) variations. We assume then that, in both
the mixing layer and the axisymmetric jet, the mean quantities, velocity,
concentration, flame surface density, and so forth, depend only on the
axial dimension, x.
With the above simplifications in mind, the model extension may be
described as follows. Within the mixing layer or jet, the reactants are
divided by the flame sheet into two streams: one containing fuel and the
other oxidizer. The reaction between fuel and oxidizer, being fast, occurs
only within the flame sheet. The slow reactions, those resulting in NO for
instance, take place both within the flame sheet and in the two fluid
streams. The chemical constitution of the two bulk fluids is determined
by four processes: (1) the volume reactions, (2) the continuous addition
to them of the products of the flame sheet reactions that occurs during
flame shortening, (3) the addition of newly-entrained fuel and oxidizer,
and (4) the removal of reactants entering the flame sheet. (In the fuel
jet, only the oxidizer fluid is continuously replenished by new reservoir
gas.) The mass rate with which flame sheet products are added to the bulk
gas is to be put equal to the flame shortening rate multiplied by the flame
mass per unit area. This latter quantity, as well as the composition, is
given by the strained flame structure calculation. The division of these
products between fuel and oxidizer streams is a required input to the model.
The calculations of Sections VI and VII and experimental data will serve as
guides in choosing this parameter. If the reaction rates can be divided
into two groups, one fast and the other slow relative to mixing rates, then
to a good approximation, the bulk fluids are homogeneous and the reactions
can be described by standard kinetic codes modified to allow for the above
described addition and removal of reactants.
41
-------
With the fuel and oxidizer gas composition known and the strain rate
determined by the local fluid mechanics, the strained flame computation
can be made as before. What is needed, therefore, are simultaneous, coupled
computations of the reactions within the bulk fluid, the strained flame
reactions, and the flame shortening rate.
The equation governing the flame surface area is the same as that in
the original model. For the mixing layer of width y, the expression is
U (y E) = a e Z y - X Z/^ + V2 E/K2 y
This equation, the stream tube, and the strained flame equations are to be
solved simultaneously for the variation of the chemical properties with x.
With regard to the strained flame calculation, the strain rate and
stream compositions vary slowly compared to the flow times within the
flame. The calculation, therefore, may be viewed as a quasi-steady one
with the properties within the flame changing instantaneously with change
in the bulk fluid state. It may not be necessary to make this computation
at each axial station. The results described in Section V suggest the
possibility of interpolation between a few such stations.
As the above described calculation proceeds step by step in the axial
direction, the change in flux of each chemical species in the mixing layer,
for example, is given by the equation
in which (flux)., is the total flux of species i, (°°) and (-°°) refer to the
fuel and oxidizer streams, and the fi's are the total production rates in
the bulk streams. The other symbols are defined in Figure 15 or have the
meanings given in the earlier sections.
It is recognized that this brief sketch leaves many details unclear.
Some have been omitted to simplify the discussion, but others remain to
be worked out. It is hoped, however, that the physical picture on which
the model is to be constructed has been adequately described.
42
-------
U-,, fuel
IL oxidizer
Figure 5. The two-dimensional mixing layer
43
-------
XII. CONCLUDING REMARKS
As was implied by the remarks at the end of Section VII, more experi-
mental data is needed on the production of nitric oxide in methane-air
flames, particularly of different scale, before a critical assessment can
be made of the present analysis. The theoretical results already obtained,
however, suggest the usefulness of a further application and development
of the model.
44
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XIII. REFERENCES
1. Marble, F. E. and Broadwell, J. E., "The Coherent Flame Model for
Turbulent Chemical Reactions," Project SQUID Technical Report No.
TRW-9-PU, January 1977.
2. Batchelor, G. K., "The Effect of Homogeneous Turbulence on Material
Lines and Surfaces," Proc. Roy. Soc. A213, p. 349.
3. Hawthorne, W. R., Weddell, D. S., and Hottel, H. C., "Mixing and
Combustion in Turbulent Gas Jets," Proceedings Third Symposium on
Combustion, Flame, and Explosion Phenomena, p. 266 (1948).
Williams and Wilkens Co., Baltimore, Md.
4. Brown, G. and Roshko, A., "On Density Effects and Large Structure in
Turbulent Mixing Layers," J. Fluid Mech. 64, pp. 775-816.
5. Konrad, J. H., "An Experimental Investigation of Mixing in Two-
Dimensional Turbulent Shear Flows with Applications to Diffusion-
Limited Chemical Reactions." Project SQUID, Purdue Univ., Indiana,
Technical Report CIT-8-PU.
6. Dimotakis, P. ••• private communication.
45
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XIV. APPENDIX A
LAMINAR DIFFUSION FLAMES WITH FAST CHEMICAL KINETICS
The reactant consumption rates enter into the problem in the form v^
and v?, the volumes of fuel and oxidizer consumed per unit flame area,
respectively, and play roles both in the species conservation relations and
in the equations describing the flame surface density. These quantities
are assumed to be determined locally by the flame structure and to depend
only upon local quantities; in particular, for the diffusion flame, they
are determined by fuel and oxidizer concentrations and a local fluid mech-
anical property. As shall be illustrated, they may be determined analytic-
ally when the kinetics are fast. The important point to keep in mind is
that the entire flame structure and chemical kinetics are coupled with the
field analysis rather weakly, so that the consideration of complex kinetics
complicates only the local flame structure and not the formulation of the
flowfield and the flame distribution,
As the first example, consider the diffusion flame with rapid kinetics;
in this approximation the reaction takes place at a surface of infinitesimal
thickness. Utilizing the coordinates x and y to signify distances parallel
to and normal to the flame surface, supposed to lie along y = 0, the fuel
and oxidizer concentrations satisfy the equations
9K-1 9K-i 9K-I la/ ^Ki \
9-r + u^r + v¥y-=p3npD977
OtCn 9Ko 9l
-------
(A4)
/Dt
and taking
v(t) = A^ (A5)
the species conservation equations are reduced to a similarity form and
become
d K_.
= 0 (A6)
where i = 1, 2 for fuel or oxidizer, respectively. This pair of differ-
ential equations is required to satisfy the conditions that the fuel and
oxidizer mass fractions take on the values K-, (°°) and K^C-00) at y = +«>
and y = -°°, respectively and that the mass flux to the diffusion flame,
y = 0, supplies fuel and oxidizer in the stoichiometric ratio. This latter
condition is explicitly
(0,t)
= f (A7)
dKo
-pD —• (0,t)
where f is the known, constant stoichiometric fuel-oxidizer ratio. It is
not difficult to show that the appropriate solution is
erf (f - A) + erf (A)
Kl " KV ' 1 + erf (A)
and
(A8)
erf (- f + A) - erf (A)
/ \ V *- ' /AQ\
K2 " K2(-°°) 1 - erf (A) (A9)
satisfying one differential Equation (A6) and the boundary conditions at
y = +00. The stoichiometry condition (Equation A7) then determines the
characteristic value A, and it is a matter of calculation to show that this
47
-------
gives the result
,H [1 - erf (A)]
- f (A10)
K;2 (--)[! + erf (A)]
Now K, (^)/k-?(-^) is the imposed fuel-oxidizer ratio of the problem, and
the quotient of this with the stoichiometric fuel-oxidizer ratio
KT (»)/K9(-~)
-! - f - =4, (All)
is frequently called the equivalence ratio. Thus, the value of A becomes
A -
This quantity defines the value of the transverse gas velocity
(A13)
which is required to keep the flame stationary at the x-axis. With this
solution, the values of the reactant volume consumption rates may be cal-
culated as
(A14)
and
This result exhibits the intuitively clear result that the consumption
rates decrease with increasing time, because the diffusion layers that
supply the reactants grow thicker with time. The equivalence ratio, which
determines the value of A, is known because K, (°°) and K?(-°°) are equal to
the values remote from the turbulent flame since the diffusion zone thick-
ness is assumed small in comparison. with flame front spacing. At any point
within the turbulent flame, therefore, the reactant consumption rates are
known in terms of the time t elapsed since the formation of the flame.
48
-------
In discussing the turbulent flame structure it was emphasized that the
turbulent motions tend to extend the flame surface and that the significant
part of this extension consists of strain rate in the plane of the flame.
Now, if the flame is aligned with the x-axis and the remaining axes chosen
so that straining rate is along the x-axis, the resulting strained diffusion
flame is also directly soluble. In the particular instance when the
straining rate in the fluid is large, i.e., where
e = ~ IA16)
is large and constant, Equations (Al) through (A3) take the form
9l
-------
flame is likewise given by Equations (A8) and (A9) together with A evaluated
by Equation (A12), but with r, as defined in Equation (A19). The corres-
ponding volume consumption rates of reactants are easily calculated
and
v2 =«2(-) V^ U + l]e-" (A23)
Thus, so far as the local flame structure is concerned, the features
relevant to the turbulent flame calculation are determined by the local
strain rate.
50
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X. APPENDIX B
TEMPERATURE AND CONCENTRATION PROFILES FOR STRAINED LAMINAR
METHANE-AIR FLAMES
51
-------
10
l/l
c
O
U
ro
s_
cu
'o
0 u
2.0
1.6
1.2
0.8
CO
I
o
0.4
y, cm
Ffgure B-l. Temperature, CH^, 02, and H20 distributions
in strained laminar methane-air flame;
e = 600 sec'1, < = < = °'75>
52
-------
Ul
CO
CM
O
X
01
O
O
fO
S-
O)
'o
E
t-D
O
X
o
(/I
c
o
•r—
4-> CO
O O
(O i—
-------
c:
o
o
to
s-
cu
o
o
o
X
o
00
o
X
C\J
:E
o
X
-------
1.2
1.0 -
-------
c:
o
o
E
O1
O
X
CM
CM
O
ro
o
X
o
HO
0.1
0.2
0.3
y, cm
0.4
0.5
0.6
Figure B-5. HO, H02, and N02 distributions in strained laminar
methane-air flame; e = 600 sec'1, K, = K? = 0.75.
56
-------
CJ1
CM_
I/)
c
o
to
S-
OJ
'o
0.4 -
Figure B-6. CH3 and H2 distributions in strained laminar
methane-air flame, e = 600 sec"1, KI = <„ = 0.75,
-------
o
o
c
o
u
1X3
OJ
o
CD
X
1C
o
un
o
X
o
0.5
0.6
Figure B-7. CRN, CH20, and NCO distributions in strained
laminar methane-air flame; e = 600 sec'1,
K, = K2 = 0.75.
58
-------
on
to
sz
o
O
(O
S-
II)
'o
X
o
nr
o
ro
O
0.1
0.2
Figure B-8. H and CHO distributions in strained laminar
methane- air flame; e = 600 sec
"1
, K, = K~ ~ 0.75,
-------
10
x 6
en
O
OJ 4
O
1.0 2.0
y, cm
3.0
2.0
CO
I
o
0.8
0.4
Figure B-9. Temperature, CH4, 02, and H20 distributions
in strained laminar methane-air flame;
e = 50 sec'1, K, = K? = 0.75.
60
-------
LO
O
X
O
12
10
l/l
o
+->
8
x
o
en
o
NO
1.0
2.0
, cm
Figure B-10. 0, N, and NO distributions in strained laminar
methane-air flame; e = 50 sec
~1
Ki =
61
-------
12
10
C.J
o
c
o
o
(13
S-
O)
o
CO,
I
1.0 2.0
y,cm
3.0
Figure B-ll. C02 and CO distributions in strained laminar
methane-air flame; e = 50 sec"1, K-, = K^ = 0.75.
62
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TECHNICAL REPORT DATA .
(Please read Instructions on the reverse before completing!
1. REPORT NO.
EPA-600/7-80-018
2.
3. RECIPIENT'S ACCESSION NO.
[. TITLE AND SUBTITLE
A Theoretical Analysis of Nitric Oxide Production in
a Methane/Air Turbulent Diffusion Flame
5 REPORT DATE
January 1980
6. PERFORMING ORGANIZATION CODE
. AUTHOR(S)
3. PERFORMING ORGANIZATION REPORT NO
Frank E. Marble (California Institute of Technology)
and James E. Broadwell
9. PERFORMING ORGANIZATION NAME AND ADDRESS
TRW Defense and Space Systems Group
One Space Park
Redondo Beach, California 90278
10. PROGRAM ELEMENT NO.
INE829
11. CONTRACT/GRANT NO.
68-02-2613
12. SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
Industrial Environmental Research Laboratory
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final: 1/78 - 4/79
14. SPONSORING AGENCY CODE
EPA/600/13
15. SUPPLEMENTARY NOTES JERL-RTP project officer is W.S. Lanier, Mail Drop 65, 919/
541-2432.
16. ABSTRACT
The report gives results of a theoretical analysis of nitric oxide production in a
methane/air turbulent diffusion flame. In the coherent flame model used, the chemi-
cal reactions take place in laminar flame elements which are lengthened by the tur-
bulent fluid motion and shortened when adjacent flame segments consume intervening
reactant. The rates with which methane and air are consumed and nitric oxide gene-
rated in the strained laminar flame are computed numerically in an independent cal-
culation. The model predicts nitric oxide levels of approximately 80 ppm at tne enc
of the flame generated by a 30. 5 cm (1 ft) diameter jet of methane issuing at 3050
cm/sec (100 ft/sec). This level varies directly with the fuel jet diameter and inver-
sely with the jet velocity. A possibly important nitric oxide production mechanism.
neglected in the analysis, can be treated in a proposed extension to the model.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
D.IDENTIFIERS'OPEN ENDED TERMS
COSATi Field Group
Pollution
ombustion
Nitrogen Oxide (NO)
Turbulence
oherence
Shear Flow
Diffusion Flames
Mathematical Models
Methane
Pollution Control
Stationary Sources
NOx Control
Coherent Structures
Methane/Air Flames
13B
2 IB
07B
20D
12A
07C
3. DISTRIBUTION STATEMENT
Release to Public
19. SECURITY CLASS (This Report)
Unclassified
!1. NO. OF PAGES
69
20 SECURITY CLASS (This pagei
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
63
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