7/24/2019 Westbrook, C. K. - Computation of Adiabatic Flame Temperatures and Other Thermodynamics Quantities
1/8
COMPUT TION OF DI B TIC FL ME TEMPER TURES
ND OTHER THERMODYN MIC QU NTITIES
Charles
K
Westbrook
Lawrence Livermore National Laboratory
Livermore California USA
Abst ract
equilibrium final temperature
and
fuel-oxidizer
mixture
both theoretical and practical importance
Both
are
needed
in order
pollutants such
oxides of nitrogen
NO
x
,
and
the final
is necessary to
predict
the
efficiency and
heating value
a g i v ~ n
c o m b u ~ t o r Knowledge
of the
ways
varlOUS radlcal and stable
intermediate
as
operating conditions
changed can
also suggest system
to
improve
overall performance.
this
paper, the methods
available
for
chemical equilibrium conditions
such
adiabatic flame temperatures will be
The dependence of
computed
values
on
be
Examples of equilibrium flame
and compositions for mixtures of
and importance in natural gas
be
presented.
OF
CHEMICAL
EQUILIBRIUM have
been
important
parts
of
many
combustion
chemical engineering
studies. In
recent
have
been
a number of
different
which have
significantly improved
reliability, speed, and cost of
In particular, the
which are used
have
extended and
made
more accurate, and new
have
been
derived
make
such calculations much easier and
less
time. Some of the
most
powerful
for
computing equilibrium
even available in
versions
which
execute on now-familiar microcomputers
and
143
The
subject of chemical equilibrium is a
v a ~ t
and
complex
scientific
discipline,
one
WhlCh has been examined in
a multitude of
books
and
research papers. This
brief
paper cannot
attempt to present
any
survey
of this
entire
field. However,
there are several narrower
issues
which
occur frequently
in practical
analyses of chemical equilibrium of combustion
systems. A situation
which
often
arises
is
that
of needing to
predict
the
final
product
temperature
and
chemical composition
which
should
be
expected
from
the oxidation
of
a
given mixture of fuel and oxidizer.
In
the
discussions to follow, the techniques and
theory will be illustrated using
either methane
CH4)
or natural
gas as
the fuel
and
molecular
oxygen
as the oxidizer, but the
m e t h ~ d s
are
very
general and
can
directly
be
applled to other fuels, oxidizers and to
multiphase systems.
In many practical
systems, combustion of
f ~ e l s such as natural gas takes place under
elther
constant pressure or effectively
constant
volume
conditions.
An example
of
constant pressure combustion is a
common
home
furnace or
gas
stove, both
of
which
are
open
to
the atmosphere.
In
contrast, although the
volume
of the combustion
chamber
in
an
internal
combustion engine is not constant throughout
the
entire
engine cycle, the combustion
of
nearly all of the
fuel-air
mixture takes place
over a ~ e r y few degrees of crankangle motion,
so
the lnternal combustion engine
can
often be
treated as a constant
volume
system. For
hydrocarbon
fuels,
constant
volume
conditions
provide higher product temperatures than
constant pressure combustion. Differences
in
equilibrium temperature
and
pressure
can result
in
differences
in
equilibrium species
compositions as well. Therefore, analysis of
these two conditions, constant pressure
and
constant volume,
can
provide information
which
can
be
applied to the great majority of
combustion environments of
practical
concern.
7/24/2019 Westbrook, C. K. - Computation of Adiabatic Flame Temperatures and Other Thermodynamics Quantities
2/8
COMPUTER
MODELS ND
SOME
THERMODYNAMICS
During the combustion of a fuel such as
methane in air, the reactants are initially at
some
reasonably low temperature, often room
temperature, prior to reaction. Burning these
reactants converts them to products and
liberates substantial amounts of heat, raising
the temperature of the products.
The
dominant
product species are water
and
carbon dioxide.
If these
were
the only products of methane or
natural
gas
combustion, then the evaluation of
the final temperature and composition of the
products would be a trivial exercise. However,
at
the temperatures common to combustion
environments,
substantial amounts
of other
species also
exist,
such as carbon monoxide
CO), trace species such as OH,
H02,
and
atoms including Hand 0. Furthermore, some of
the molecular nitrogen N2) present in the
air is converted to various oxides of nitrogen
NO
x
), including NO,
N02, and N20.
The
relative amounts of these minor constituents
vary with temperature,
and
some
of the heat of
reaction is consumed
in
creating
these
trace
species. To compute the proper final
equilibrium temperature of a
particular
reactive gas mixture, the existence
and
relative
concentrations of these minor species
must be correctly taken into account.
As
an
example
of the
size
of the error which can
result
from ignoring these minor products,
if
methane
and air are
burned
at room temperature
and atmospheric pressure, the correct product
temperature is found to
be
about
2220K,
or
about 3537 of. If all of the product species
except C02 and H20 are neglected, the
computed
product temperature is
2320K,
or about
3717
of. The most important minor constituent
is usually CO, and if
this
species is included
in the
final
product distribution, then the
computed product temperature
becomes 2250K,
or
3591
of. Errors of as much as lOOK in product
temperature
can
lead to serious problems in
prediction of rates of production of
NO
x
,
which are extremely
sensitive
to temperature,
so
i t
is quite important to retain these minor
chemical constituents in the calculations of
equilibrium composition.
If it
is assumed that all of the heat of
reaction
goes
into the product gases
and none
of the heat is allowed to escape
from
the
closed system, then the combustion takes place
adiabatically. The final equilibrium product
temperature is properly designated
as
the
adiabatic flame temperature Tad only if the
process takes place under constant pressure
conditions.
The final
temperature at constant
volume will
be somewhat
higher than Tad.
This simple
thermodynamic fact
can best be seen
by thinking of constant pressure combustion as
a sequence of constant volume combustion
followed
by an
expansion of the product gases
back to the initial pressure. The product
gases will cool as they do work on their
surroundings to arrive at the initial pressure.
44
Oxides of nitrogen
NO
x
) present a
particularly important class of minor p r o d ~ t
species in combustion. Because these specles,
especially
NO
and N02,
combine
with
atmospheric water to produce strong acids,
emissions of
NO
x
from
combustion systems must
be minimized wherever possible. At combustion
temperatures, most trace species such as
OH,
0,
H02,
and
many
others arrive at their
equilibrium concentrations very
rapidly,
often
over time scales of the order of milliseconds
or less.
In
contrast, production of
NO and
N02 takes place over much longer time scales,
often
as long
as tenths of seconds.
Furthermore,
as
noted above,
this
time scale
for the production of
NO
x
is very sensitive
to temperature. Therefore, if the temperature
history of the product gases can be controlled
to minimize the residence time at high
temperatures, then there will
be insufficient
time for complete equilibration of the products
to produce significant amounts of
NO
x
.
An
equilibrium computation of the species
concentrations represents in
this
sense a
worst case of
NO
x
produ ct i on.
From
a chemical
kinetic
point of view,
equilibrium is achieved in a combustion system
when the net rate of every chemical reaction is
exactly zero. This
means
that the forward rate
of each reaction is equal to the reverse rate
of the reaction. From a thermodynamic point of
view, it also means that the appropriate work
function has a minimum value. For constant
pressure conditions,
this
is the Gibbs free
energy G, and for constant volume conditions it
is the Helmholtz free energy
F. The two
are
re 1ated by
F G P V
where P is the pressure
and
V is the specific
volume. The Gibbs free energy is defined
as
G
~ j n j
where
l U ~ is the chemical potential for
species j
and
nj
refers
to the mole fraction
of species j in the given
gas
mixture.
The
mathematical process of minimizing G
is
formally constrained by requiring that the
number of
atoms
of each type in the mixture
e.g. C, 0,
N,
and
H for hydrocarbon
combustion)
be
conserved. In
most
of the
methods for minimizing the free energy, these
constraints
are included by means of the
technique of Lagrange
multipliers [1,2J.
It is also possible in
principle
to
compute
this equilibrium state
in
a
dynamic
fashion, following the time evolution of the
system of chemical species in
which
the kinetic
rate equations have been included. If the
rates
of the forward and reverse reactions have
been properly defined in terms of the relevant
equilibrium constants, then eventually the
interconnected set of species concentrations
will settle down to its own equilibrium at the
7/24/2019 Westbrook, C. K. - Computation of Adiabatic Flame Temperatures and Other Thermodynamics Quantities
3/8
final
temperature. This technique,
employed
to examine chemical kinetics
[3J,
differential
equations for the
as
they
in time. The rate of each reaction is a
total density, and
that
in
reactions.
This, however, is
a competitive computational technique for
equilibrium
states,
since it requires
more computer time and
more
numerical methods for its
However, at equilibrium
rates
of the forward and reverse
reactions
so it is
possible to replace the
equations for the time rate of
of each species concentration with a set
simple
algebraic
equations for those species
makes
the equations much
to solve,
and
by using equilibrium
ctions the formal conservation of each type
atom
is
made
automatic. This approach
has
used to calculate equilibria in large
[4J.
The
primary thermochemical data
used by
and
specific
heats of
each
of the chemical
The specific enthalpy H
for
is
generally expressed
as
a
specific
p
T
W T) W(Tref)
SCpdT
I,-.z.\-
Tref is conventionally 298K or
room
For
virtually
every chemical
interest to combustion, the specific
A
common
procedure, particularly
when
a rough estimate of the
final
state is
is
to
assume
that the
specific
heat
is
so
the above
integral can be
replaced
the e x ~ e s s i o n
C
p
T
-
Tref .
C
p
dT
T ..
this approach
can result in errors
of
lOOK
in the product temperatures.
The
'
most
useful compilations of heats of
and specific heats as functions of
JANAF
tables [5J, which
assembled
and
revised over the past
years. Other sources of thermochemical data
have been
recently by
Burcat [6J.
For
use in
fit to
in
temperature, so
that
integrations of
specific
heat over
can be carried
out very
Errors introduced into the
by
this
fitting
procedure
negligibly
small in
most
cases.
145
Of
course, the
entire
approach is only
as
good as the basic thermodynamic data being
used. For the small molecules
treated
by the
JANAF
tables, the
most
recent
tables
are quite
reliable, although earlier
JANAF
versions for
some species
have changed
considerably as
more
experimental analysis
has been
done. For
larger
molecules
and
for
many
radical species,
specific heats
and
heats of formation are not
as
well
known
and
uncertainties
in equilibrium
product distributions and temperatures are
certainly
possible.
For
organizations
and
individuals
who wish
to carry out the types of calculations already
described
and
illustrated below), many
of
the
best computer
programs
for equilbrium
conditions are available to the general public.
The original program of Gordon and McBride [lJ
from NASA and
the
program
developed
by Reynolds
[2J
at
Stanford
can be
purchased
and
can
even
be run on
personal computers, while an
adaptation of Reynolds' program
has
been
integrated into the widely
used
CHEMKIN [7J
family of combustion
modeling
programs
developed
by
Sandia Laboratories.
Those
interested
in obtaining
such
programs should
contact the authors of these
codes
for further
information.
To
summarize this
section of the
topic,
computations of chemical equilibrium require
two
major elements.
The
first of these is a
numerical
model which can
solve the relevant
equilibrium equations.
Such
a
code
can solve
time-dependent kinetics equations, algebraic
equilibrium
relations,
or
use
a free energy
minimization technique.
In principle, any
method should arrive at the same solution,
although the computer time
and
cost
requirements of
each
technique can be widely
different, and the most
commonly used
technique
is
that
of free energy minimization.
The
second
major
element
is
the basic thermodynamic
data set
which
is
used by
the
computer
model,
and all common
models
use basically
the
same
thermodynamic data set. It is quite important
to
include in the equilibrium computation
all
chemical species
which may be
present
at
equilibrium
above
a
certain
threshold
level,
since omission of species
such
as O and
H
can result in significant errors in
computed
equilibrium temperatures.
SOME
EXAMPLES
Let us consider first a reference or
baseline
set
of
initial
conditions to
illustrate
the types of information
which can
be
obtained
by
thermodynamic
equilibrium
models. This is a mixture
of
natural
gas and
air, in
which
the fuel is defined to consist of
90CH4, 5 H 6 ~ with a 5impurity
level of
N2.
The
air
is a mixture of
oxygen
and nitrogen, with
N2 02 =
3.78.
All
of
the gases are
assumed
to be
initially
at
atmospheric pressure and 60F.
Using
Reynolds'
7/24/2019 Westbrook, C. K. - Computation of Adiabatic Flame Temperatures and Other Thermodynamics Quantities
4/8
method [2J
as
adapted by Kee and co-workers at
Sandia [7J, the adiabatic flame temperature is
calculated to be
3528F
at atmospheric
pressure.
The
chemical species concentrations
at this
final
state
are
listed in
order of
concentration in the first column
below
~ S ~ p e ~ c ~ i ~ e ~ s
__
~ u ~ l ~ l _ l ~ i ~ s ~ t ~ S ~ h ~ o ~ r ~ t _ l ~ i ~ s ~ t
__
C ~ O
____ H 2 _
N2
H2
0
C02
CO
02
H2
OH
NO
H
o
Temp 0 K
0.71212
0.18010
0.08629
0.00859
0.00439
0.00342
0.00265
0.00189
0.00036
0.00020
2220
0.71812
0.18633
0.09555
2318
0.71450 0.71378
0.18539 0.18177
0.08499 0.08631
0.01008 0.00863
0.00504 0.00605
0.00343
2252 2240
with
all
other species
such as H02, N02,
N20, and H202 less
than 1
ppm mole
fraction
of
less
than 1.0 x 10-6
.
If
we
repeat the
same calculation
but
assume that
the
only possible products are N2, C02, H20,
and 02, the adiabatic flame temperature is
then
found
to be
2318K
or 3713
of,
with the
composition
shown
in the second column of the
above
table. The third
column
shows
the
computed
results
when
CO
is added to
the
product list,
and
the fourth column shows the
results
when
H2 is then included.
Comparisons
of these
results
with the complete
list
in the first column above
indicate
that
the greatest improvement comes from inclusion
of
CO, which is
the most important
minor
product, but
that
the changes
resulting
from
consideration of the other species are
st i l l
quite noticeable.
Using the
above
reference case for
comparison,
we can now
address the influences
of variations in three common operating
parameters on adiabatic flame temperature and
equilibrium composition, with NO noted in
particular. These basic parameters are the
fuel-air equivalence ratio the amount of
excess oxygen, and the preheat temperature of
the air. We will treat these quantities as
independent parameters, although in practice
they are often used together to fit a
particular
need
for a given combustor.
EQUIVALENCE
RATIO
-
The
most important quantity
that changes with is the adiabatic flame
temperature, reaching a maximum
value for
equivalence ratios close to stoichiometric
= 1 .
Because
most combustion reactions
depend
quite
strongly on temperature
[3J,
the
overall combustion
rate
and
related
quantities
like
the flame speed also are
found
to reach
maximum
values for = 1. Similarly, the
equilibrium concentrations of most
of the
radical species generally are
greatest
near
stoichiometric.
146
Using
the
same
computer
model as used
earlier,
we can
compute
the adiabatic flam:
temperature and product species concentratlons
for methane-air mixtures initially at
atmospheric temperature and pressure.
The
results for the temperature and some
illustrative
species concentrations are
shown
in Figures 1-3.
2 5 0 0 ~ ~
~
:> 2000
Q
s
1500
~ 1000
s
Q
0..
S
Q
E-
...-4
(l)
~
2800
(l)
2600
oJ
C\1
~
2400
l)
~
S
(l)
2200
E-
2000
0 20 40
60
80 100
Percent
02
Figure 4
Adiabatic flame temperature
as
a function of
oxygen
content in the oxygen-nitrogen mixture
for stoichiometric methane-oxygen mixtures
initially at
room temperature and pressure.
0 4
-r-----------------
0.3
C
0
oJ
c l
C\1
~
0.2
to-.
(l)
...-4
0
0.1
CO
CO
2
~
7 . ~
/
---
,, . __ . . . . H2
0.0 - - - - ~ - - - . . - - - - - . - - - - - . . . - - - - - I
o
20
40 60 80
100
Percent
02
Figure 5
Selected species concentrations at equilibrium,
plotted
as
functions of
oxygen
content in the
oxygen-nitrogen mixture.
In the plots of species equilibrium
concentrations, note
that
the
minor
constituents such
as
CO, H2 and 02 all rise
with
02
enrichment.
As
the percentage of
02
is increased, the equilibria should shift
towards
C02 and H20
and
away
from
CO
and
H2, based simply on the Law of
Mass
Action.
However, since the temperature of the products
is
also increasing, and the position of the
equilibrium is even
more
sensitive to
148
C
0
oJ
c l
C\1
~
to-.
(l)
. . .-4
0
~
0.10
0.08
0.06
0.04
0.02
OH
.....
/
.
--.,.:.
/ ...... H
,
-
/
: ,
/
/
(
0.00 -+---......,....=-----.----...----.------4
o
20 40 60 80
100
Percent 02
Figure 6
Selected species concentrations
at
equilibrium,
plotted
as functions of oxygen content in the
oxygen-nitrogen mixture.
temperature than to concentration, the overall
result provides more partially oxidized
products than completely oxidized species.
Note also
that
the NO level increases
rapidly with
02
enrichment, due exclusively
to the increased product temperature. In this
case, the attainment of these higher
equilibrium NO concentrations will also be
faster
as
the
02
enrichment is increased.
The sharp drop in NO level at very large
02
enrichment levels
is
a
result
of the
way
the
computations
were
done; at
100 02
in the
oxidizer,
there is
no
molecular nitrogen in the
lIairll, so no
NO
or NO
x
can
be produced .
AIR
PREHEAT -
In many practical
systems the
oxidizing air stream
can be
heated prior to
mixing
with the fuel to provide higher product
temperatures. A series of adiabatic flame
temperature and equilibrium computations was
carried
out to examine this
issue. The
pressure
was
held fixed
at
atmospheric, and the
air preheat temperature was varied from zero
(i.e. room
temperature air) to 2000F.
The
results
of these computations are
plotted
in
Figures 7-9. The
final
temperature increases
steadily with air preheat temperature. This
could be expected because for these mixtures
the ratio of air to fuel is constant at 5.76/1,
so
the
reactant
mixtures are predominantly
air.
Because of the steady temperature rise, the
levels of all of the incompletely oxidized
species and
radicals
also increase with air
preheat temperature. Furthermore, the NO level
increases steadly with air preheat
as well, and
again the production
rate
of NO and
its rate
of
attaining that equilibrium level will also
increase with
air
preheat temperature.
7/24/2019 Westbrook, C. K. - Computation of Adiabatic Flame Temperatures and Other Thermodynamics Quantities
7/8
4 5 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~
4000
3500
3 0 0 0 ~ - - - - - - . _ - - - - - - ~ - - - - - - ~ - - - - ~
o 500 1000
1500 2000
Air
temperature
degrees F
F gu re 7
as
a function of
preheat temperature for stoichiometric
at atmospheric
pressure
initially at room temperature.
0.20 - ------------------------------.....
0.15
0.10
..................................
0.05
_ CO
0.
00
- - - - - - - ~ - - - - - - ~ - - - - - - - r - - - - - - - - I
o
500
1000
1500 2000
Air
temperature
degrees F
Figure 8
at
equilibrium
as
functions of
air
preheat temperature
stoichiometric methane-air mixtures.
149
~
0
.-
oJ
C)
ro
f 4
c
Q)
.-.
0
::;E
0.020 - ------------------------------
0.015
0.010
0.005
H
NO
............
.................
0.000 - - - - - - - ~ - - - - - - , . . . . . - - - - - - - - - r - - - - - - - I
o
500 1000
1500 2000
Air temperature
degrees
F
Fi gu re 9
Selected species concentrations at equilibrium
plotted
as
functions of air preheat temperature
in stoichiometric methane-air mixtures.
SUMMARY
These examples
of
adiabatic flame
temperature and chemical equilibrium
composition calculations serve to illustrate
several major points. First the results of
such
parameter studies provide a significant
amount
of useful information concerning
combustion properties of
practical
systems. In
industries
where the accurate prediction of
product species concentrations pollutant
emission levels
and
operating temperatures is
important there is a continuing need for
computational capabilities of this type.
The specific examples used here are not
revolutionary; in fact those industries which
need such information already use data of this
type every day
in normal
operation.
However,
the present paper points out the need for
continual updating of the thermochemical data
base
upon which
these computations depend.
Another point
made
was that the equilibrium
data alone are
insufficient to predict some
combustor properties such
as NO
x
emission
since the rate of attaining equilibrium is
another
variable
that may not
be
related to the
equilibrium properties themselves.
Finally the current state-of-the-art
computer programs for
calculating adiabatic
flame temperatures equilibrium compositions
and
many other useful thermochemical quantities
that
this paper
has
not
had
the time or space
to
discuss are easily obtained are generally
quite simple
and
convenient to use
and can
even
be
run on personal computers and other
7/24/2019 Westbrook, C. K. - Computation of Adiabatic Flame Temperatures and Other Thermodynamics Quantities
8/8
common
computer systems in general use. The
acquisition and
implementation of these
programs by industrial research, and other
organizations
is
highly
recommended.
ACKNOWLEDGMENT
This
work was
supported
by
the
Gas
Research
Institute
and was performed under the
auspices of the U.S. Department of Energy
by
the
Lawrence
Livermore National Laboratory
under
contract No.
W-7405-ENG-48.
REFERENCES
1. Gordon, S. and McBride,
B.J.
Computer
Program
for Calculation of
Complex Chemical
Equilibrium Compositions,
Rocket
Performance,
Incident and Reflected Shocks, and
Chapman-Jouguet Detonations , National
Aeronautics
and Space
Administration report
NASA SP-273, 1971.
2. Reynolds,
W.C.,
STANJAN
Interactive
Computer Programs
for
Chemical
Equilbirium
Analysis, Stanford University, 1981.
3. Westbrook, C.K., and Dryer, F.L.
Chemical
Kinetics Modeling of Hydrocarbon Combustion ,
Progress in
Energy and Combustion
Science 10, 1
(1984). ---
4. Meintjes, K.,
and Morgan,
A.P.,
Performance of Algorithms for Calculating the
Equilibrium Composition of a Mixture of Gases ,
J. Computational Physics 60, 219 (1985).
5.
JANAF
Thermochemical Tables, U.S. National
Bureau of Standards NSRDS NBS 37 and
supplements,
D.R. Stull
and H. Prophet, eds.
6. Burcat, A., Thermochemi ca
1 Data
for
Combustion
Calculations , in
Combustion
Chemistry, W.C. Gardiner, Jr. ed.
Springer-Verlag, New York, 1984.
7.
Kee,
R.J.
Miller
J.A. and
Jefferson
T H
CHE MKI
N: A Genera l-Purpose,
Problem-Independent, Transportable, Fortran
Chemical
Kinetics
Code
Package , Sandia
National Laboratories report SAND80-8003, 1980.
8. Baulch, D.L., Drysdale, D.O., Horne, D.G.,
and
Lloyd, A.C., Evaluated Kinetic Data for
High Temperature Reactions, vol. 2 ,
Butterworths,
London,
1973.
15