Westbrook, C. K. - Computation of Adiabatic Flame Temperatures and Other Thermodynamics Quantities

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.


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


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'


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.


7/8

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


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,

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Westbrook, C. K. - Computation of Adiabatic Flame Temperatures and Other Thermodynamics Quantities

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Transcript of Westbrook, C. K. - Computation of Adiabatic Flame Temperatures and Other Thermodynamics Quantities