EPA-600/7-90-003
January 1990
A MODEL OF TURBULENT DIFFUSION FLAMES
AND NITRIC OXIDE. GENERATION
by:
J.E. Broadwell
(TRW Space and Technology Group)
and
T.J. Tyson
C.J. Kau
Energy and Environmental Research Corporation
18 Mason
Irvine. CA 92718
EPA Contract 68-02-3633
EPA Project Officer
W. Steven Lanier
Air and Energy Engineering Research Laboratory
Research Triangle Park. NC 27711
Air and Energy Engineering Research Laboratory
Office of Research and Development
U. S. Environmental Protection Agency
Research Triangle Park, NC 27711
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BIBLIOGRAPHIC INFORMATION
PB90-155557
Report Nos; none
Title: Model of Turbulent Diffusion Flames and Nitric Oxide Generation.
Date: Jan 90
Authors: J. E. Broadwell, T. J. Tyson, and C. J. Kau.
Performing Organization: TRW Space and Technology Group, Redondo Beach, CA.-'"'Energy
and Environmental Research Corp., Irvine, CA.
Performing Organization Report Nos: EPA/600/7-90/003
Sponsoring Organization: "Environmental Protection Agency, Research Triangle Park,
NC. Air and Energy Engineering Research Lab.
Contract Nos: EPA-68-02-3633
Supplementary Notes: Prepared in cooperation with Energy and Environmental Research
Corp., Irvine, CA. Sponsored by Environmental Protection Agency, Research Triangle
Park, NC. Air and Energy Engineering Research Lab.
NTIS Field/Group Codes: 81A, 68A
Price: PC A04/MF A01
Availabi1ity: Available from the National Technical Information Service,
Springfield, VA. 22161
Number of Pages: 53p
Keywords: '''Diffusion flames, '"'Turbulence, "Nitrogen oxide(NO), '"'Mixing, Combustion,
Combustion products, Mathematical models, Flame propagation, Reaction kinetics,
Chemical reactions, Experimental data.
Abstract: A new view is described of mixing and chemical reactions in turbulent
fuel jets discharging into. air. Review of available fundamental data from jet
flames leads to the idea that mixing begins with a large scale, inviscid,
intertwining of entrained air and fuel throughout the jet. Significant molecular
mixing is delayed until the end of a cascade to the Kolmogorov scale. A simple
mathematical model incorporating these ideas is presented. The model predicts a
Reynolds number dependence for the nitric oxide formation rate that is in good
agreement with measurements in both methane and hydrogen jets burning in air. These
mathematical model concepts have been incorporated into a simplified computer
program capable of treating the detailed chemical kinetics of a gas flame. The
model has been used to predict NO formation in H2/air and CH4/air flames and
results compare favorably with experimental data.
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NOTICE
This document has been reviewed in accordance with
U.S. Environmental Protection Agency policy and
approved for publication. Mention of trade names
or commercial products does not constitute endorse-
ment or recommendation for use.
ii
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ABSTRACT
—A new view is described of mixing and chemical reactions in turbulent
fuel jets discharging into air. Review of available fundamental data from jet
flames leads to the idea that mixing begins with a large scale, inviscid,
intertwining of entrained air and fuel throughout the jet. Significant
molecular mixing is delayed until the end of a cascade to the Kolmogorov
scale. A simple mathematical model incorporating these ideas is presented.
This model predicts a Reynolds number dependence for the nitric oxide
formation rate that is in good agreement with measurements in both methane and
hydrogen jets burning in air. These mathematical model concepts have been
incorporated into a simplified computer program capable of treating the
detailed chemical kinetics of a gas flame. The model has been used to predict
NO formation in Hj/air and ChU/air flames and results compare favorably with
experimental data.
¦ii'i'i ;
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TABLE OF CONTENTS
Section Page
Abstract /ii i-.
List of Figures rvi r
List of Tables
Acknowledgements SWM.i
1.0 Introduction 1-1
PART I - THEORETICAL DEVELOPMENT (J.E. Broadwell) 1-1
1.1 Jet Mixing and Flame Length 1-1
1.2 The Jet Structure 1-9
1.3 The Treatment of Slow Chemical Reactions: 1-10
Nitric Oxide Generation in Fuel Jets in Air
PART II - NUMERICAL ANALYSIS (T.J. Tyson. C.J. Kau, J.E. Broadwell) .. 11 -1
11.1 Introduction 11 -1
11.2 Model Development 11 -1
11.3 Numerical Procedures and Heat Transfer Model 11-12
11.4 Sample Calculation and Sensitivity of Model Parameters ... 11-13
11.5 Comparison with Experimental Data and 11-15
Discussion of Results
11.6 Concluding Remarks 11-24
References 11-26
\v "
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LIST OF FIGURES
Fi qure Page
1-1 Effect of Normality Ratio and Reynolds Number 1-2
on Liquid Visual Jet Length
1-2 Effect of Normality Ratio on Visual Liquid Jet 1-4
Length for Nozzles of Various Shapes
1-3 Sketch of Turbulent Jet Entrainment 1-5
1-4 Dependence of Turbulent Flame Length on (p0/p«) 1/2<> 1-8
1-5 Instantaneous Temperature Profiles Across a I -11
Two-Oimensional Heated Jet at x/D = 4.5
1-5 Sketch of Turbulent Jet Reaction Model 1-13
2-1 Sketch of Turbulent Jet Entrainment 11 - 3
2-2 Flame Length vs. /m^\ fp 11 - 5
\ro)Jt
2-3 Schematic of Reactor Model 11-10
2-4 Sensitivity of p(a = 0.7) 11-14
2-5 Sensitivity of Radiative Heat Transfer 11-16
Parameter a(p = 10)
2-6 Effect of Radiative Heat Transfer on Flame Core 11-17
Temperature History
2-7 Comparison of Model Calculated NO vs. Measurements 11-18
(Fr = 6 x 10-5) for H2/Air Flames
2-8 Predicted E/t0 versus Re0 for H2/Still Air 11-19
Flames (at Constant Froude No., Fr - 6 x 105)
2-9 Overall NO Formation in Flame-sheet Zone and 11-21
Overall NO Destruction in Flame-core Zone
2-10 Emission Index versus Re0 (H2/Air Flame) 11-23
v i,1'--
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LIST OF TABLES
Table Page
2-1 Flame Length Data 11 - 6
2-2 Comparison of Predicted and Measured Emission Indices ... 11-25
for CH4/Air Turbulent Diffusion Flame
vii
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ACKNOWLEDGEMENTS
The authors are pleased to acknowledge the contributions of many
discussions with Frank Marble to the development of the ideas reported here.
One of the authors (J.E.B.) is grateful to his colleagues of the Fluid
Mechanics Group at the Cal ifornia^Institute of Technology for generously
sharing their new insights into turbulent flows. The early experimental
investigation of jet structure by Paul Dimotakis was especially valuable.
Special thanks are also due to R.E. Breidenthal who was an unofficial
col 1aborator.
v'iii-1
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1.0
INTRODUCTION
One of the central focuses of EPA's combustion research program has been
development of combustion control technologies to limit formation of nitrogen
oxides (N0X) from fossil fuel fired flames. The Fundamental Combustion
Research (FCR) program has served to help advance understanding of critical
combustion processes controlling N0X formation. The present study was
conducted to develop an improved understanding of the coupling between
fundamental fluid mechanics and chemical processes in turbulent diffusion
flame. Specifically, the study focused on one of the simplest flow
configurations: a jet of methane fuel discharging into ambient air.
Many practical stationary combustion processes use diffusion type flames
where the combustion rate is controlled by mixing of fuel and air. A typical
characteristic of low N0X burners for pulverized coal fired utility boilers is
that the fuel/air mixing rates are retarded providing an elongated flame. The
present FCR project does not address the practical problems associated with
utility boiler burners but instead focuses on better understanding the
critical fuel/air mixing process in flames.
Classical fluid mechanics provides the view of turbulent flow as an
enhancement to the viscous processes where, for example, mixing rates are
describable through the product of a mean velocity gradient and a "turbulent"
diffusivity. Similarly, shear stress could be described by the product of
mean velocity gradients and a "turbulent" viscosity. These classical flow
theory notions have been extremely effective in describing bulk flow processes
such as flow of fluid through pipes or over aircraft wings. Recent
experiments and turbulent theory developments (see for example Brown and
Roshko, 1974 and Roshko, 1977) are leading to a new view of the basic
processes involved in turbulent shear flow. This new view has evolved from
recognition of the existence and importance of large-scale organized structure
in turbulent shear flows. The large-scale structures are seen to engulf fluid
from either side and distribute it across the layer. The mixing of fluids on
a molecular scale occurs at the interface formed between alternating layers of
jet and reservoir fluid. If these alternating layers represent fuel and
oxidizer, the local combustion environment is radically different than would
be predicted by classical turbulent mixing theories.
1-1
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f
The current FCR study was conducted in two parallel parts and the report
is also organized in that way. In Part I, a mathematical model has been
developed while in Part II. this model is extended and adapted to a computer
simulation. Both parts of the study and report make comparison of predictions
with available experimental results.
1-2
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PART I THEORETICAL DEVELOPMENT (J.E. Broadwell)
1.1 Jet Mixing and Flame Length
Information providing important insights into turbulent mixing process
is available from a variety of sources including a particularly significant
study by Weddell (1941). A major review of the early turbulent flame studies
is provided by Hottel (1953) in the Fourth Combustion Symposium. Hottel's
review covers the work of Weddell, who studied liquid jets consisting of an
alkali solution marked with phenophthalein discharging into an acid reservoir.
The jet. which is initially red, becomes colorless when it entrains and mixes
with sufficient acid. Weddell states that the flow strikingly resembles a
turbulent flame. In the experiments, the reservoir-jet normality ratio was
varied between two and ten. This ratio, R, is the mass of reservoir fluid
required to cause the color to vanish from a unit mass of jet fluid when the
two are mixed homogeneously. For fixed values of R, the distance, L. over
which the color was visible was interpreted as "flame length" and presented by
Hottel as a function of jet Reynolds number, Re0. Figure 1-1 is replotted
from Hottel's paper.* Here Re0 - pU0d0/n. where d0 and U0 are the initial jet
diameter and velocity respectively, p the density and p. the viscosity.
One of the noteworthy features in Figure 1-1 is that at sufficiently
high Reynolds number the flame length, measured in jet diameters, becomes
independent of Reynolds number. An explanation for this observation was
proposed some time ago by Broadwell (see Witte et. al.,1974). When the marked
jet disappears every element of jet fluid has molecularly mixed with at least
R parts of reservoir fluid. The required reservoir fluid is entrained into
the jet at some upstream location. Subsequent mixing of the jet and reservoir
fluids occurs by intertwining on an ever decreasing seale--cascade--to the
Kolmogorov scale X0. These intertwining motions are i nvi sci d and reach X0
after time, x
-------
p0 uo Vy ' 10-3
"igure 1-1.
Effect of Normality Ratio and Reynolds Number
on Liquid Visual Jet Length, from Hottel (1953).
1-2
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Xo ~ (v3/c)l/4 ~ (v3d/u3)1/4
where v is the kinematic viscosity, \i/p, and z is the kinetic energy
dissipation rate ~ u3/d (see, for instance. Tennekes and Lumley, 1972). The
time for diffusion to occur across this micro scale. x%. is given by
- VO ~ Sc/Redi/2 (d/u). (1)
where D is the molecular diffusion coefficient, Sc is the Schmidt number
<^/pD). and Rea - pud/ji. Therefore, the ratio. T. of the diffusion time to
the large scale break-down time is
T ~ tx/Td ~ Sc/Redi'2 - Se/Re0i/2 (2)
since jet Reynolds number does not vary with axial distance, x.
Equation (2) above can be used to explain the observations from Figure 1-
1. The equation shows that at high Reynolds number the diffusion time is
short compared to the entrainment-breakdown time and consequently the mixing
time is limited by the slower, Reynolds number independent process. Stated
another way. negligible additional distance is required for molecular mixing
to be completed after the jet and reservoir fluids have intertwined and
cascaded to Xq. Finally, when the chemical reaction rate is fast relative to
the diffusion rate (as it is in the phenophthal ein reaction), the "flame
length" will also be independent of Re0. Equation (2) also predicts that for
high Reynolds number the flame length is independent of Schmidt number.
Evidence on this point is discussed later.
Weddel1 combined the high Reynolds number data in Figure 1-1 with data
for nozzles of other size and shape and presented it in the form of Figure 1-2
to show that "flame length" varies almost linearly with normality ratio, R.
The argument developed above for Reynolds number independence explains this
result as well. Figure 1-3 is a sketch of a circular jet schematically
indicating entrainment, which is a constant value per unit length. The jet in
Weddell's experiments disappears when every element of jet fluid is
molecularly mixed with R units of reservoir fluid. In Figure 1-3, suppose
that reservoir fluid enters the jet between the origin and x0 and that
1-3
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R
Figure 1-2.
Effect of Normality Ratio on Visual Liquid Jet Length for Nozzles
of Various Shapes from Hottel (1953).
1-4
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Figure 1-3. Sketch of Turbulent Jet Entrainment.
1-5
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molecular mixing is complete at x. Since the quantity of entrained fluid
rises linearly, the amount molecularly mixed will also increase linearly, at a
slower rate, if x is linear with x0. That this is so follows immediately from
the above discussion: since the breakdown time is proportioned to d/u, the
required distance is proportional to d and hence to x0. Then since negligible
additional distance is needed for the molecular mixing, the flame length
varies 1inearly with R.
Hottel (1953) also reviewed the flame length data for various gaseous
fuel jets in air and compared it with the theoretical work of Hawthorne.
Weddell, and Hottel (1949) and others. The above analysis may be applied to
these jets if the difference in fuel and air density and the effect of heat
release on density can be accounted for. An approach to accounting for these
effects is outlined next.
For current purposes, attention will be restricted to the high Reynold's
number jets and to situations where the buoyancy force is small compared with
the jet momentum. Initial consideration will be given to non-reacting jets
whose density differs from that of air (the reservoir fluid). Ricou and
Spalding (1961) examined this case experimentally and found that, at
sufficiently high Reynolds number, entrainment is independent of Reynolds
number and is linear with axial distance, x. Their experiments show also that
the effect of the jet-air density difference on entrainment can be expressed
in the form
mt/m0 = Ci(pa/p0)i/2 x/d0. (3)
in which mt and m0 are the total and initial jet mass flow respectively, pa and
p0 are the air and fuel density and Ci is a constant equal to 0.32. The
authors point out that the form of equation (3) confirms a result form
dimensional analysis that the entrainment is determined by the jet momentum
(see Section II.2 in the current report). For the current study a similar
argument is made that, as an approximation at least, the form of equation (3)
would also apply when there is reaction between the fuel jet and air. The
previous analysis of chemically reacting liquid jets can be combined with the
entrainment estimate of equation (3) to develop a prediction of the flame
length for gas fuels in air.
1-6
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First it should be noted that equation (3) is applicable only when x/d0
is large and the entrained air flow is correspondingly large relative to the
fuel flow. m0. Then mt is approximately the air flow rate, ma. The fuel will
be completely consumed at the axial station where a stoichiometric amount of
air has been mol ecu! arly mixed with every element of fuel. The argument given
above that the molecular mixing rate is controlled by the entrainment rate
allows mt/m0 in equation (3) to be set equal to the air-fuel stoichiometric
ratio, 4>. and leads to the prediction that flame length, L. will vary
according to
in which C2 is a new constant (which will be smaller than Cj). As before the
fuel-air reaction rate is assumed to be rapid relative to mixing.
Hottel (1953) collected flame length data for several fuels (hence
different values of <{>). These data are presented in Figure 1-4 and are seen
to be approximately linear with (p0/pa)1/2(j> in accord with equation (4). Also
shown in Figure 1-4 is the average "flame length" data for liquid jets given
earlier in Figure 1-2. Note that the normality ration R in Weddell's
experiments has the same effective meaning as 0. The previous analysis
leading to equation (2) had led to the prediction that at high Reynolds number
flame length should be independent of Schmidt number. This is confirmed in
Figure 1-4 where little difference is seen between liquid and gaseous "flame
length" even though the Schmidt number ratio is approximately 1000.
Hottel makes no direct comparison between the liquid and gaseous results
but presents a correlation of the fuel data with the analysis of Hawthorne,
Weddell, and Hottel (1949). In that analysis, the flame length is correlated
with the jet width at which the stoichiometric amount of air is entrained.
The jet spread angle is assumed constant and equal for all fuels, and chemical
reactions are assumed to occur simultaneously with entrainment. The resulting
equation for flame length is
L/d0 - (l/c2) (Po/Pa)1^.
(4)
L - s
j - 5.3 (<|> + 1)
Tf 1
Ma 1/2
a T0 ($ + 1) +
(0 + 1) M0
1-7
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Figure 1-4. i
Dependence of Turbulent Flame Length on (Pq/P.) $•
Data from Hottel (1953).
1-8
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where Tf if the adiabatic combustion temperature of a stoichiometric mixture.
T0 the nozzle gas temperature, a the moles of reactants per mole of products
in a stoichiometric mixture, Ma ana M0 are the molecular weights of air and
nozzle gas respectively, and s the distance to the jet break point. It is
seen that for $ large compared to unity and with the assumption the Tf and a
do not vary widely, this expression reduces in form to that derived above.
That the two analyses should be in approximate agreement is not surprising
once it has been established that molecular mixing rate is linear in axial
distance as is the entrainment.
Before summarizing the above results, it is important to clarify an
aspect of the mixing process that has not been discussed. Following the
entrainment of a particular volume of reservoir fluid into the jet and during
the subsequent cascade in scales to the Kolmogorov scale. X0, molecular
diffusion is occurring at the interface between the two fluids. Hence some
molecular mixing takes place before X0 is reached. The end of the flame,
however, is reached when every part of the jet fluid mixes with the required
amount from the reservoir. Thus x\ in equation (1) is the time required for
the completion of mixing after the Kolmogorov scale is reached.
The analysis developed above, with its satisfactory agreement with
experiment, may be summarized as follows. In both gaseous and liquid jets at
sufficiently high Reynolds number, the distance needed to mix, at the
molecular level, a given amount of reservoir fluid with the jet fluid is
independent of Reynolds number. The length is determined by the large scale
entrainment rate. The "flame lengths" for gases and liquids are. therefore,
approximately the same, i.e., the flame length is independent of Schmidt
number. Finally the flame length depends linearly on the parameter
. a consequence of the assumption that the jet entrainment rate is
determined solely by the jet momentum.
1.2 The Jet Structure
The picture of the mixing process in jets that was developed in the
preceding section emerged in an attempt to understand the data in Figure 1-4.
The emphasis in that section was on the dependence of mixing on the various
parameters. However, the magnitude of the mixing rate implied by the data.
1-9
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together with the parameter dependence has further significance. Note from
Figure 1-2 that at 20 diameters from the nozzle exit, for instance, every
element of jet fluid is mixed with at least two parts from the reservoir.
Hence large quantities of the entrained fluid reach the jet center near the
nozzle exit. This fact, viewed in the light of recent photographic evidence
obtained by P.E. Dimotakis (1981). suggests that reservoir fluid is carried
into the jet in large scale coherent structures and that the breakdown to
small scales begins from this initial state. A natural idealization of these
processes is that there is no radial variation of the reservoir-jet mixture
ratio in the mixed fluid. In this view, the usual bell-shaped mean
concentration profiles arise only from a radial variation in the ratio of
mixed fluid to pure reservoir fluid.
For two-dimensional jets, the measurement of Uberoi and Singh (1975)
provide strong support for this picture. They projected hot wires across a
slightly heated jet to obtain virtually instantaneous temperature
distributions, examples of which are given in Figure 1-5. Within the jet
there is seen to be little lateral temperature variation. These authors
interpret their results to mean the lateral position of the jet is fluctuating
and show that a superposition of the instantaneous profiles yields the well-
known bell-shaped curves.
The inference to be drawn from these data and from the discussion in
Section 1.1 is that mixing in turbulent jets does not occur through a gradient
diffusion process. Instead the two fluids intermix uniformly on a large scale
and reach a molecularly mixed state as has already been described. The same
view of mixing in shear layers has been put forward by Broadwell and
Breidenthal (1981).
I.3 The Treatment of Slow Chemical Reactions:
Nitric Oxide Generation in Fuel Jets in Air
The first step in developing a model framework for treating nitric oxide
generation in jet is recognition that the mixing rate inferred from Figure 1-4
applies regardless of chemical reactions that may take place. That is, for
example, in the case of a fuel of stoichiometric ratio, , equal to ten and
flame length 100 d0, all the fuel elements are mixed at upstream stations with
either air or partial combustion products at a linear rate. This mixing
occurs at a rate, of course, such that the flame would have ended at 20 d0 had
I -10
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y/D
-igure 1-5.
Instantaneous Temperature Profiles Across a Two-Dimensional
Heated Jet at x/D = 45. "ean Velocity Axis 0.305 CM/SEC.
From Uberoi and-Singh (1975)-. . - .
1-11
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<(» been equal to two. The basis for this assertion is that a difference in
molecular diffusion coefficients between reactants and products have no
influence on the mixing rate. An explicit expression for the air addition
rate will be given later.
The analysis and discussion to this point has dealt with situations in
which the chemical reaction rates are fast relative to mixing, and in which,
therefore the reaction rate is controlled by the entrainment rate. This
approximation is appropriate for the energy releasing reactions in fuel jets
but may not be appropriate for the slower reactions which generate nitric
oxide by thermal fixation. The inclusion of these reactions, therefore,
necessitates a refinement in the model.
In the model as formulated so far, the fast reactions take place in the
molecular diffusion zones-strained laminar flames (see Marbel and Broadwell
1977, 1980)-during the cascade to the Kolmogorov scale, X<,, and as the mixture
is homogenized by diffusion at this scale. In both situations the fuel and
oxidizer react at the stoichiometric ratio. The slow reactions occur during
these processes as well but continue, at rates set by the chemical kinetics,
after the mixture has been homogenized.
These ideas can best be described in Lagrangian frame, i.e., by
following a unit mass of fuel after it leaves the nozzle and mixes with air at
the known rate.
At any given time, the fuel mass and the air entrained up to that time
are undergoing the cascade to X0 or have reached this scale. It will be
argued later in this report that the nature of the cascade is such that
relatively little flame is generated until just before Xo is attained and the
mixture is homogenized. If this is the case, the reactions in these two
processes may. as an approximation, be treated together and the time that the
reactants remain at the stoichoimetric ratio is limited by the diffusion time
x\, derived in Section 1.1.
The model is conveniently described with the help of the sketch in
Figure 1-6. The reactors A and B together contain a unit mass of fuel to
which air is added at the rate ma. The reactions that occur at stoichiometric
conditions take place in reactor B; slow reactions proceed in both A and B.
1-12
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B
m
a
-fp
*
Figure 1-6. Sketch of Turbulent Jet Reaction Model.
1-13
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It is assumed, as a tentative approximation, that the salient feature of the
reactions in B is that they occur at stoichiometric conditions and that the
details of the strained flame structure and of the homogenizing diffusion
process are unimportant. A natural implementation of this approximation would
be to take reactor B to be a well-stirred chemical reactor and reactor A as
plug flow reactor. This concept will be developed in more detail in Part II
of this report. However, it should be noted that a well-stirred reactor with
appropriate residence time is a close simulation of the reaction zone of a
strained flame.
As implied above, a critical parameter in the model is the residence
time in reactor B, tb- Analysis presented earlier argues that this would be %\
if most of the flame sheet were generated just before the scale of
concentration variation were reduced to XQ. A detailed examination of the
nature of the buildup in time of flame sheet between a given quantity of
entrained air and the fuel-product mixture indicates that this may be so. The
approach was formulated by R.E. Breidenthal (see Broadwell and Breidenthal
1981) for this process in turbulent shear layers, and modified by V.A.
Kulkarny (private communication).
The idea is that the interface surface, S, per unit volume, being
inversely proportional to the scale of the distortions, X. caused by the
turbulence can be estimated from the scale variation with time. It is assumed
that the Kolmogorov description of the cascade in scales applies to the jet
mixing, at least at the small scales. The procedure is as follows.
The basic concept [see. for instance, Landau and Lifshitz (1959)] is
that for scales down to X0, the motion is inviscid and the kinetic energy
flux, e, is, constant, i.e., that
e ~ v3/X. ~ u3/d for X0 < X < d.
This statement alone is not sufficient to determine the dependence of X on the
time, t, and further progress must rely solely on dimensional arguments. To
quote Landau and Lifshitz "v?v may also be regarded as the velocity of
turbulent eddies of size X." Thus if it is assumed that in one "turn over
time" the eddy gives up a large fraction of its energy, the time spent at Xi
i s
1-14
-------
At ~ Xx/\li ~ Xi/el/3 Xl/3 ~ X2/3/el/3-
Now a further decision is required concerning the new scale generated. V.A.
Kulkarny argues that, because the process is self similar, the change in scale
should be scaled with X. and thus the memory of the initial scale lost.
Therefore
^ X2/3
dt « *1 A d*
el/3 X (5)
where k, is a constant.* Integration of equation (5) and use of the
expression for e give an estimate for the time to reach a given scale X,
starting at scale d.
i(X)=k2 d/u [l - (X/dF3]
and since X,, - *3 d/Re3/4, the time td to reach X<, is
Td = k2 d/u (l - k3/Rc01/'2) = k2 d/u, for Re0 »1
where k's are constants. Thus in accord with the discussion in Section 1.1,
at high Re0 the time to reach Xo is independent of Re0. With S ~ 1/X the
growth in flame sheet area per unit volume can be expressed as
S- 1/d (l - t/xd)-3/2,
which shows the anticipated rapid increase in S when t near td.
Since the diffusion at X0 follows immediately, it is reasonable to
assume that the fuel and oxidizer are mixed stoichiometrically only for the
time required by the diffusion process, ~ td/Re01/z (when the Schmidt number
* Here, for convenience, a continuous change in scale is assumed. The
actual situation may better be described by jumps between discreet scales, but
only qualitative results are required at this stage of model development.
1-15
-------
is unity). Thus the residence time in reactor B, Tb, is taken proportional to
(d/u)Re0-i/2. If radiation is neglected, the mixture is at the adiabatic glame
temperature for this length of time also. Since the nitric oxide producing
reaction rates depend strongly on temperature it would appear to follow
immediately that the nitric oxide level would scale with
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PART II NUMERICAL ANALYSIS (T.J. Tyson. C.J. Kau, J.E. Broadwel1 )
11.1 Introduction
This second part of the report develops a detailed numerical model of a
simple gaseous turbulent diffusion flame with associated nitric oxide
formation. The basic underlying concepts of the model were presented in Part
I and are summarized as follows:
• Macroscopic entrainment of a turbulent jet is independent of
Reynolds number and is linearly proportional to the axial
(mainflow direction) distance.
• The macroscopically entrained fluid is mixed uniformly with the
jet fluid so that there is no macroscopic variation in radial
direction.
• The macroscopically entrained fluid is engulfed into the jet in
large coherent structures; subsequently, breakdown to small scale
structures is through inviscid, instable motion.
• The majority of the interface surfaces are generated just before
the structures reach the Kolmogorov scale.
«
• Combustion takes place in the interface surfaces (or flame sheets)
stoichiometrically at the Kolmogorov scale, and the interface
surfaces are homogenized by molecular diffusion at this scale.
The mathematical model postulated here consists of simple chemical
reactors arranged in a manner to account for the basic features listed above
as well as the detailed chemical kinetics of combustion reactions and NO
formation. A simple flame radiation model in incorporated to account for
radiative energy loose which can have a strong impact on temperature and NO
formati on.
11.2 Model Development
In this section, the differential equation describing the proposed model
are put in a form suitable for numerical computations. The basic turbulent
11 -1
-------
jet mixing/reaction model developed in Part I will be partially redeveloped to
assist in empirically evaluating critical model constants. The basic flow
configuration to be modeled is a circular jet of fuel issuing into a reservoir
of stationary air.
For the basic flow condition being modeled, the conservation of momentum
flux requires that
J = ^ Pm U m dm — TtPo U0 d0
4
where subscripts "m" and "o" refer to the local mean and initial jet
properties respectively and all other terms are as defined earlier in Part I.
The configuration is illustrated in Figure 2-1. The above equation can be
rearranged to provide an expression for the local mean main flow velocity Um
in terms of initial density p0 and initial velocity U0 by:
Um =(-&-)1/2 (uQdp|
\ dm I (6)
By definition, the local mass flux per unit initial jet mass flux can be
written as
rc* 11-Pm^m dm
PoU0dn
Eliminating Um from the above equation by using Eqn. (6) gives
\m„/ * P° M d0 ' (7)
As in Part I. the jet spreads linearly at approximately a constant angle,
except for within a few diameters from the nozzle exit and is independent of
the type of jet fluid and physical conditions. The jet radius is in linear
proportion to the axial distance, i.e.,
dm - 2k2X (8)
where k2 is the tangent of half jet angle.
11-2
-------
11 - 3
-------
Substituting Eqn. (8) into Eqn. (7) becomes:
(9)
The turbulent entrainment law derived by Ricou and Spalding (1961) recommended
k2 - 0.16. CCi from equation (3) equals 2 times k2.] The axial distance where
turbulent-jet-entrained stoichiometric amount of air can be evaluated from
Eqn. (9) by simply setting (mt/m0) to be stoichiometric ratio, i.e.:
Flame length is the indication of the axial distance where the
stoichiometric amount of air is microscopically mixed with the fuel when
combustion reactions are fast and when combustion is also mixing controlled.
The flame length data for various gaseous fuel jets in air from the classic
work of Hawthorne, Weddel1 and Hottel (1949) and the recent work of Bilger and
Beck (1974) and Peter and Donnerback (1981) are plotted in Figure 2-2. The
physical parameters of these flames are also given in Table 2-1.
Experimentally, the flame length here is referred to as the analytical flame
length which represents the axial distance of the point where 99 percent of
the combustion is completed (based on sample gas). As can be seen from Figure
2-2, almost all the points fall on a straight line which has a slope of 9.217.
The lengths [Eqn. (10) - K2 - 0.16] of the macroscopic entrainment of
stoichiometric amount of air for various flames are also plotted as the solid
line in Figure 2-1. The plot suggests a linear proportionality of flame length
to
and a constant ratio of the microscopic mixing rate to the macroscopic
entrainment rate, i.e..
(10)
hll| .ns:
m0/Sl V Pm
(^tinicro = k 1 (rot)m;
lacro
13-4
-------
T
7T
O CH4 Peters and Donnerhack (1984)' 217
C3H8 ) /
*-2H2 ( Hawthorne, et al (1949) /
300
O Hz
X CO
200
=k
Bilger and Beck (1974) y
A
/
/
/
/
/
V
/
/
/
u
o
>4.1 o
100
/
/
/
/
/
/
/
/
i
/
/
°/
/
¦From Eqn. (10)
3.125
Figure 2-2. Flame Length v.s.
11-5
-------
Table 2-1. Flame Length Data.
Data
No.
Fuel
Composition
d '
0
(cm)
po
P *
_o
pm
P)
V^o/st
Re **
0
p p***
(tt
w
\t\li
Lf/D
I/O****
ri.a
Source
1
Hydrogen
h2
0.476
14.45
0.138
35.047
2870
92,000
13.02
114
40.68
Hawthorne et
a 1
(1949) '
2
Hydrogen
h2
0.462
14.45
0.138
35.047
3580
158,000
13.02
126
40.68
Hawthorne et
a 1
(1949) ,
3
Propane
C3H8
0.3175
0.67
2.985
16.7
29,000
54,000
28.85
263
90.16
Hawthorne et
a 1 1
(1949,)
4
Acetylene
c2h2
0.3175
1.036
1.930
14.2
17,400
88.000
19.72
176
61.70
Hawthorne et
al
(1949)
5
Acetylene
c2 h2
0.3175
1.034
1.934
14.2
32,400
307,000
19.74
108
¦ 61.70
Hawthorne et
al
(1949)
6
CO
0.635
1.034
1.934
3.45
5,090
2,415
4.79
75.6
14.99
Hawthorne et
al
(1949)
7
Hydrogen
h2
0.635
14.45
0.138
35.047
12,270
60,000
13.02
120
40.68
Bilger & Beck (1974)
8
ch4
0.30
1.80
1.11
17.20
3,200
14,710
18.12
170
56.63
Peters et ali
A(1984)
* p ~ 0.5 p ** Reynolds No. = U /\> *** Froude No. = **** From Eqn. (10)
III Cl 00^ 0*^0
I /
-------
Numerically, Figure 2-1 gives the proportional constant, klt a value of 0.339;
that is to say. the microscopic mixing rate is approximately one-third of the
macroscopic entrainment rate. By replacing the macroscopic entrainment rate
in Eqn. (10) with the above relation, the expression for dimensionless flame
length becomes:
k=_i (hi) jw
do 2kik2 U0/St V Pm (11)
The logical extension of the above conclusion is that "the microscopic mixing
rate in linearly proportional to rPo~l X\ ' ^'e'
»Pm (d
(ull) =k3J&:U\
™o'micro vdo' (12)
where k3 - 2kxk2 - 0.1085.
The definition of velocity from Eqn. (6) gives
Um = dK = J.£o_ Updo
dt v Pm 2k2X (13)
Carrying out the integration of the above equation with respect to time
by assuming a constant mean density. pm, results in the following space-time
relati on:
JL - ( Po )1/4 / Uq 1 ^ '
d0 P"1 lk2d0' (14)
Substituting Eqn. (14) into Eqn. (12) results in an expression for microscopic
mixing in terms of time:
11-7
-------
|hl) =k3 f_eo_)l/4 fiVL)1
'micro sl)"4 Ui-J"2
d tun 0/micro & 2 \k2d0t/
1/2
(16)
(17)
where <|> is the air fuel mass ratio of the microscopic mixture. The overall
unburned raw fuel left in the flame per unit of initial raw fuel flux is
evaluated by
i.o - -MHi
'o'micro
'Pm' \k2d0; (18)
Experimental evidence discussed in Part I on turbulent jet mixing
suggests that:
• The macroscopically entrained air is first engulfed into large
scale coherent structures and is subsequently cascaded down to the
Kolmogorov scale.
• The mixing of the macroscopically entrained air is approximately
uniform across the jet in a radial direction.
11-8
-------
• The majority of the interface surfaces are generated just before
the structures have reached the Kolmogorov scale. The interface
surfaces are homogenized by molecular diffusion at this scale.
Based on the above experimental foundation, a simple turbulent diffusion
flame model is postulated which contains only two major zones: the flame
sheet and flame core. The flame sheet zone is the Kolmogorov scale structure
which contains microscopically mixed fuel and air at a stoichiometric
condition. The flame core zone consists of a mixture of the combustion
products and the as-yet unreacted fuel which are molecularly mixed. After its
formation, the flame sheet zone will subsequently conglomerate into the flame
core structure via molecular diffusion process. Numerically, the flame sheet
zone can be simulated by a well stirred reactor with proper characteristic
residence time. The flame core zone can be represented by a plug flow
reactor, which originally contains 100 percent raw fuel, and is continuously
fed by the combustion products from the flame sheet reactor. In the meantime,
the flame core zone also has to provide the flame sheet reactor with the rich
mixture which contains a sufficient amount of raw fuel for stoichiometric
combustion (corresponding to the rate of microscopic mixing of fresh air).
The idea defined above was illustrated in Figure 1-6 and is provided in
a bit more detail as Figure 2-3. Quantification of the flow to each reactor is
discussed below.
The rate of fresh air which is mixed into the flame sheet reactor can be
computed by using Eqn. (16) and the rate of raw fuel to the reactor can be
calculated by using Eqn. (17). The overall rate of mixture from the flame
core reactor, which includes raw fuel and its diluent (combustion products),
can be derived from Eqns. (16). (17). and (18), and expressed by
The proper residence time for the flame sheet reactor corresponds to the
molecular viscosity and the Kolmogoroff scale; i.e..
11-9
-------
_d_ j mfd
dt \ m
Figure 2-3. Schematic of Reactor Model.
11-10
-------
where Ci is a proportional constant and v is the kinematic viscosity. As
discussed in Part I the Kolmogorov scale is evaluated by
*=C2(£)"4
where C2 is another proportional constant. The rate of dissipation of
turbulence is estimated by
E=C3iin!
d
Thus, the characteristic residence time for the flame sheet reactor can be
expressed by
x-QC22 yi^d1/2
Um3/2 (19)
Eliminating Um by using Eqn. (12). the residence time expression becomes:
U 3/2 I \d0l (20)
2
where P 5 ^1^2 ^^3 is a major physical uncertainty of the model. Expressing
the residence time in terms of clock time by using Eqn. (14), we have
x=fl4k2(p?)
1/4
VrT (21)
Reynolds number Re is defined as
11-11
-------
Re = Uod°
v
where v is the kinetmatic viscosity of the combustion products at typical
flame temperatures. For CH4/air flames v is taken to be 1.2 cm2/sec while for
H2/air flame v is taken to be 4.0 cm2/sec. Since the typical flame
temperature is not known a prior, there is some uncertainty in v. However,
this uncertainty is minor and can be absorbed into the major certainty of the
model. p.
II-3 Numerical Procedures and Heat Transfer Model
A computer code has been developed for the turbulent diffusion flame
model which consists of a well stirred reactor and a plug flow reactor. The
code was developed by making use of EER's existing one-dimensional reactor
code. The numerical aspect of the diffusion flame model includes the
following features:
• Implicit Finite Difference: The governing species and energy
equations for both reactors are solved implicitly by backward
finite difference formulation in time-like coordinate.
• Fully Detailed Kinetic Mechanism: The FCR I kinetic set consists
of 181 elementary reactions and 36 species for CH4/air flames
which includes detailed NO and fuel nitrogen mechanisms. The
kinetic set can be reduced to 17 species and 60 elementary
reactions for H2/air flames. (See Seeker, et. al , 1985 for
details on kinetic reaction set.)
• Gas Radiative Heat Transfer: The radiative heat exchange between
radiating gas and ambient is considered. The major radiating
species in hydrocarbon flames are H2O, CO2 and CO. For small-
scale gaseous flames, an optically thin approximation, which
assumes that all the radiative energy from the flame escapes the
flame envelope should be sufficiently accurate. For large-scale
flames, self-absorption has to be considered. For modeling of
simple turbulent jet diffusion flames, the axial dimension is much
larger than the radial dimension and therefore, the radiative flux
11-12
-------
in the axial direction tends to be absorbed while the radial
radiative flux tends to escape. Richter (1974) has estimated the
radial and axial energy flux for a typical turbulent diffusion
flame by using the four-flux model. He concluded that only about
30 percent of the radiative energy radiates in the axial directon
while 70 percent of the energy radiates radially. As a first
order approximation in the current model, it is assumed that all
the radiative flux in the axial direction will be self-absorbed
while all the radiative flux in the radial direction will be lost
to the ambient.
11.4 Sample Calculation and Sensitivity of Model Parameters
As the first example, which also serves to check out the correctness of
the numerical procedures in the program, one of the CH4/air turbulent
diffusion flames measured by Bilger and Beck (1975), has been selected for
simulation. This flame is a hydrogen jet issuing from a 0.635 cm nozzle into
still air. The mean exit velocity is 19,323 cm/sec. Both hydrogen and air
are initially at 300 K and at one atmosphere Pressure. The computer code was
exercised for various values of p and the calculated NO concentration
profiles* are shown in Figure 2-4. As illustrated, p — 10 gives the best
agreement between prediction and experimental data. However, it should be
noted that the universality of p requires further analytical and numerical
studies.
In order to examine the sensitivity of the radiative heat transfer model
on NO formation, computations were made for the particular flame by
artificially varying the radiative heat transfer parameter, a, from 0.5 to
1.0. For example, a - 0.5 means 50 percent self-absorption, while a - 1.0
represents an optical-thin limit of heat loss. The results are plotted in
Figure 2-5. To illustrate the impact of the heat transfer model on
temperature, the profiles of the flame core temperature histories are plotted
in Figure 2-6. No attempt was made to compare the calculated temperature
profile against the measurements since temperature measurements give only the
* The computed NO concentration in the plots is the mean of instantan-
eous NO concentration from both reactors multiplied by kj which accounts for
the dilution effect of macroscopic air.
11-13
-------
T
200|— ( D0 = 6-35 11111
H2/Air Flame ]Uo = 19,323 cm/sec
Q.
C.
c
o
*5
t/)
*3
Re„ =12270
o
O Measurements^)
,,-n. ^ , S = 20
!5Cj— / \
/ -Ck\
7 ^ NV o
icq— ' / N\\
/ 7
/ / vs
I O /
/ / °
ss
/ /
50( / /
/ /
/ /
/ /
/ /
//
^
8 = 10
8 » 1.0
20 60 100 140
Axial Distance (X/DQ)
Figure 2-4. Sensitivity of S (a = 0.7)
11-14
-------
200
150
Q.
Q.
C
o
o
(/>
TJ
100
50
H^/Air Flame
d0= 6.35 nsn
U0 = 19,323
Re0 = 12270
cm
sec
6 = 10
a Varying
^a = 0.5
Nd sn
// NO SN
'' \ N
\ s
Va ¦ o-7 '
\ N
\
^ = 1.0
80
100 120
140
Axial Distance U/DQ)
Figure 2-5. Sensitivity of Radiative Heat Transfer Parameter a (S - 10)
T1-15 . |
-------
Non-Dimensional Axial Distance (X/DQ)
Figure 2-6. Effect of Radiative Heat Transfer on Flame Core Temperature History.
11-16
-------
mean temperature of the macroscopic mixture which contains both the gases of
the flame sheet and the flame core as well as the macromixed fresh air.
11.5 CQHipanson—with P£rim6nt& 1 Data and Discussion—of Resul ts
By setting p = 10, the model is used to predict the NO concentration of
the various H2/air flames reported by Bilger and Beck. (1975). The computed
NO histories of three flames with constant Froude numbers (accomplished by
varying U0 and d0 at the same time) are presented in Figure 2-7 and compared
against the measured centerline NO concentrations. The agreements are
encouraging. The same numerical results are also plotted differently in
Figure 2-8 in terms of emission index per unit convective time, E/x0, versus
Reynolds number of fuel jet. Re0. The convective time scales, x0, is defined
as the ratio of the nozzle diameter to the exit velocity. The emission index,
E. is defined as the NO mass flux per unit of fuel mass flux. Figure 2-8
indicates a (Re0)"1/2 dependency of E/x0 as was noted in the original paper by
Bilger and Beck. However, it should be noted that when the Froude number
(U02/gd0) is held constant, x0 and Re0 are not mutually independent. Instead,
t0 is in proportion to (Re0)i/3f since by definition
pr , Up2 a (Re0P _ (Updo/vp)" _ (Uor m
gdo (x0r (do/(vo)p(dorn <22)
Two equation can be developed for n and m,
n + m - 2
and
m - n - 1
thus requiring that
n - 1/2 and m - 3/2
If Fr is held constant, then
11-17
-------
20C-
150
a.
a.
Jf1
o
>-
c
o
£ 100
u
l/l
l/l
«3
s:
50
Model
Prediction
~
\
\
as
/
/
/
/ &/ si\
/ / \°%
7 ' V\
4 / /S >5 \
\
O
/
/
; 'V
<7/
'/
/ ' Experimental lata
^ v.
v N\ s,
\ \
X X
U£
"/
"/
///
Case/
Symbol
Do
(mm)
uo
(cm/sec'
Re
To
usee
Re0
1 ~
1.59
9,669
384
16
1,537
2 A
3.18
13,674
1087
23
4,348
3 O
5.35
19,323
3067
33
12,270
20
60
100
140
Axial Oistance U/0Q)
igure 2-7. Comparison of Model Calculated NO Against Measurements
(Fr = 6 x 10~5) for h^/Air Flames.
11-18
-------
1
100
1
10^
J I I I 1 I
5 6 7 8 9 1
10'
UA
Re - -2^2.
o
Figure 2-8. Predicted £/tq Against Re0 for H2/Sti11 Air Flames
(at Constant frcude No., ~r = fix 10$).
11-19
-------
To cxRe01/3
Furthermore, if we try to correlate E with x0 and Re0, we find that
E a a 1
Reoq (Rc0)q"p/3 (23)
Since the data in Figure 8 shows (Re)'1/2 dependency, we have:
Ett—^—a J
(Re0)1/2 (Re0)1/6 (24)
Thus the dependency of emission index (from Bilger and Beck, 1975) is actually
-1/6. However, this type of correlation is meaningless since x0 was not held
constant.
Figure 2-9 shows the overall NO formation (mole/gm of fuel) in the flame
sheet zone and flame core zone as functions of axial distance for flame No. 7
in Table 2-1 (Re0 =» 12,270). The results indicate that NO is formed
predominantly in flame sheet zone while approximately one third of that is
destroyed in fuel-rich flame core zone.
Figure 2-10 shows the calculated NO emission index over a wide range of
RE0 while holding x0 constant (i.e.. t0 - 23 usee). In the lower range of Re0
(Re0 < 2 x 104) the emission index is insensitive to the increment of Re0 due
to the following contradictory effects:
• Increasing Re reduces flamelet residence time and therefore
reduces the time duration for NO kinetics.
• Increasing Re reduces flamelet residence time which in turn
reduces flamelet radiation time and increases the flamelet
temperature. This will increase the rate of NO formation.
11-20
-------
Figure 2-9. Overall NO Formation in Flame-sheet Zone and
Overall NO Destruction in Flame-core Zone
(Re0 = 12270, UQ = 19,323 cm/sec, DQ = 6.35 mm).
11-21
-------
Figure 2-10. Emission Index Versus ReQ (H2/Air Flame).
-------
• Increasing Re reduces flamelet residence time and thus causes
higher super-equilibrium radical pool concentration which will
increase NO formation.
In the mid-Re0 range (Re0 - 2 x 104 ~ 106), the first effect is prevailing.
Thus, the emission index is proportional to the flamelet residence time which,
in turn, is proportional to (Re0)-i/2. In the very high Re0 range (Re0 > 106)
the effect of the flame sheet is negligible due to short flame sheet residence
time (x << tno). NO is predominantly formed within the flame core zone where
the equivalence ratio is close to unity (i.e., near the flame tip). The NO
formation within this range is found to be proportional to convective time
scale, but insensitive to Re0.
Experimental data for free CH4/air turbulent diffusion flames with NO
measurements are very scarce. Recently Peters and Donnerhack (1981) have
reported their experimental data on NO in CH4/air flames. However, the
experiment was performed in a co-flowing fuel-air system. The overall
equivalence ratio of all flames was maintained at a constant value of 0.63 by
varying air and CH4 flow rates. The size of flame tube (external air tube)
was fixed while the fuel nozzle diameter was varied. These flames are
confined in nature, which limits the free supply of air for microscopic
entrainment. The pressure gradient which occurs as a result of confinement
could cause flow recirculation which in turn would affect the patterns of
macroscopic entrainment and microscopic mixing. Techniques for treating
confined flow are beyond the scope of this report. For the purpose of this
report, we have selected only the flames from Peters and Donnerhack's data
which are least affected by confinement. These flames are from the smallest
fuel tubes which also have the highest fuel jet velocities and the lowest air
velocities. The parameters of these flames and the emission indices are
presented along with the results from the numerical model in Table 2. The
predicted emission indices are within 15 percent of the measured value.
11.6 Concluding Remarks
A model has been formulated that describes, in a simple way, chemical
reactions in turbulent fuel jets discharging into air. The model deals with
both fast and slow reactions and is applicable therefore to nitric oxide
generation processes.
11-23
-------
Table 2-2. Comparison of Predicted and Measured Emission Indices for
CH^/Air Turbulent Diffusion Flame.
Flame No.
21
22
23
24
¦?0 (on)
0.3
0.3
0.35
0.35
UQ (cm/sec)
2,070.
1,720.
1,780.
o
IX)
CO
r—1
Reo
3,200.
2,630.
3,200.
3,340.
Fr
14,710.
10,000.
9,335.
10,000.
Lf/°o
170.
157.
163.
171.
cal. >< 11)3
1.07
1.14
' 1.03
1.02
; 11-24 j
-------
A basic aspect of the model is that molecular mixing which precedes the
reaction does riot take place as a consequence of the "turbulent diffusion" of
the reactants towards each other down concentration gradients. Instead fuel
and air are entrained throughout the jet, intertwine in large scale inviscid
motions, and mix molecularly at the interfaces and as the mixture is
homogenized at the Kolmogorov scale. This view of the mixing process leads to
a model in which the state of the fuel-air mixture is independent of the
radial coordinate. The usual bell-shaped distributions arise solely from the
radial variation in the ratio of pure air to the fuel-air mixture.
Based on these concepts a numerical model has been developed and used to
predict NO formation in simple turbulent diffusion flames. The results
compare favorably against experimental data for both H2/air and CH4/air flames
of various Reynolds numbers. The model results were found to be sensitive to
a parameter p which influences the calculated residence time in the model's
flame-sheet reactor zone. Encouraging results were achieved by empirically
setting this parameter equal to 10. The model predictions are also sensitive
to radiation heat transfer but a first order approximation, applicable to the
simple flame types being models provided satisfactory comparisons to
experimental data.
11-25
-------
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Destruction in Flames. Progress in Energy and Combustion Science. Chigier,
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Bilger. R.W. and Beck. R.E. 1975 Further experiments on turbulent jet
diffusion flames. Fifteenth Symposium (International) on Combustion, p. 541.
The Combustion Institute.
Broadwell, J.E. and Breidenthal, R.E. 1981 A simple model of mixing and
chemical reaction in a turbulent shear layer. GAICIT Report FM81-10,
California Institute of Technology.
Brown, G.L. and Roshko, A. 1974 On density effects and large structure in
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11-26
-------
Peters, N. and Donnerhack, S. 1981 Structure and similarity of nitric oxide
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11-27
------- |