LEWIS NUMBER EFFECTS ON THE STRUCTURE AND EXTINCTION
OF DIFFUSION FLAMES DUE TO STRAIN
A. Liñan
Escuela Técnica Superior de Ingenieros Aeronáuticos
Universidad Politécnica de Madrid
SPAIN
In turbulent diffusion flames chemical reaction and mixing occur simultaneously in
thin strained and distorted laminar mixing layers separating fuel from the oxidizer.
If the reaction can be modelled in terms of an irreversible reaction
F + n 0
2
—> Products
the relative importance of chemical production and transportation terms in the con-
servaron equations is measured by the Damkohler number, or ratio of the character-
istic mixing time t
m
and the characteristic chemical time t
c
. A typical mixing time
is the inverse of the straining rate y, while t
c
~' is proportional to the frequency
factor of the reaction times the Arrhenius exponential exp
(-E/RT)
involving the
ratio of the activation energy of the reaction E to the thermal energy TR. Combus-
tión reactions are exothermic and with large valúes of the ratio E/RT, and, as a
result we find in the flow-fiéld regions of low temperature, where the Damkohler
number is very small and, therefore, the chemical reaction is frozen. These regions
coexist with regions of high temperature where because the reaction is so fast, one
of the reactants is depleted, so that either the local concentration of the fuel or
of the oxidizer must be zero.
Thus in flows with combustión reactions we find regions in the flowf
iel
dwithout fuel
separated from regions without oxidizer by thin diffusion flames, where the reactions
take place by a process controlled by the rate of diffusion of the reactants towards
this surface. These equilibrium regions can be separated from regions of frozen
flows,
where the reactants coexist, by thin premixed flames that move relative to
the unburned mixture leaving behind the reactant that was in excess. See Liñan and
Crespo
(1976).
For very reactive mixtures the regions of frozen flow cover a small fraction of the
flowfields. The diffusion flames are anchored at the injector rim and the chemical
334
reaction Is almost everewhere diffusion controlled. However in many cases the fíame
is blown completely off the flowfield or lifted from the injector rim, so that a
región of mixing without significant effects of the chemical reaction appears,
bounded from the downstream región of diffusion controlled combustión by premixed
flames.
A criterium for the validity of the diffusion controlled assumption, and the condi-
tions leading to local extinction due to fíame strain can only be obtained from an
analysis of the structure of the thin reaction zones. See Liñan (197*0 and Peters
(1980) and Peters and Williams
(1980).
These analyses show that due to finite rate
effects the temperature in the reaction zone is lower than the asymptotic fíame
temperature valué associated with the diffusion controlled limit, and that it
decreases with decreasing valúes of the Damkohler number; then due to the large
sensitivity of the reaction rate with temperature no diffusion controlled combustión
is possible below an extinction valué of the Damkohler number.
With the assumption of inifinite reaction rates and if the Lewis number of the
reacting species is unity, the passive scalar approach, see for example Bilger
(1976),
can be used to reduce the problem of describing the reactions in turbulent
flows in unpremixed systems to a turbulent mixing problem; thís being specially the
case if the effects of density changes due to chemical heat reléase are neglected,
although they may be very important in increasing the coherence of the large eddy
structures in turbulent combustión.
The purpose of the following is to ¡Ilústrate the effect of the Lewis number of the
fuel in the structures of a strained laminar mixing layer between adjacent eddies
of fuel and oxidizer. If a constant positive strain is maintained for some time,
the mixing layer will reach a steady state described by the following conservation
equations:
vZY,
+ DY„+ vB Y Y
r
e~
T
a
/T
= 0 0)
' oZ oZZ o F
Y
Z í
c7
1 L~
1
D Y
r
,, + B Y Y
c
e"
T
a
/T
= 0 (2)
FZ FZZ o F
y Z T
z
+ D T
zz
-(Q/c
p
) B Y
Q
Y
p
e"
T
a
/T
= 0 (3)
to be solved with the boundary conditions
Y = Y
n
- 1 = T - T = 0 for Z ->--« (í()
o F o
Y = Y = Y
r
= T - T = 0 for z -> » (5)
o 0°° F o
335
giving
the
mass fractions
Y , Y of the
oxidizer
and
fuel,
and the
temperature
T, in
terms
of the
distance
to the
reference stagnatíon plañe.
We
consider,
for
simplicity,
that
the
flow field
is not
modified
due to
changes
in
density,
so
that
the
velocity
in
the Z
direction
is
(-yZ), where
y is the
straining rate.
We
also consider
the
diffusivity
of the
oxidizer
D to be
constant
and
equal
to the
thermal diffusivity,
so that
the
Lewis number
for the
oxygen
is
taken equal
to
unity, which
is not an
unreasonable assumption;
the
Lewis number
of the
fuel
is L
different from unity.
The reaction between fuel
and
oxidizer
is
modelled
by an
Arrhenius reaction
of
frequency factor
B,
activation temperature
T
a
=
E/R, mass stoichiometr ic coefficient
oxidizer/fuel
v, and
heat reléase
Q per
un it mass
of
fuel.
The
temperature
of
both
approaching streams
is T.
This problem
was
analyzed
for
large valúes
of the
nondimensional activation energy
T
/T , in the
particular case
L=1 by
this author
(197*0,
and the
results were later
used
by
Williams (1975)
to
describe
the
structure
of
diffusion flamelets
in
turbu-
lent combustión.
The
analysis showed
the
existence
of a
premixed fíame regime,
in
addition
to the
diffusion controlled regime that appears
for
large valúes
of the
Damkohler number B/v. Because
in
most applications
the
ratio
Y /v is
small,
1
rr
o*»
typically 0.06,
it
turns
out
that extinction condition occur
in the
premixed fíame
regime that
we
shal
1
describe,
in the
following,
for L 4 1.
For large valúes
of the
nondimensional activation energy
T /T the
reaction
in the
3
3
' a o
premixed fíame regime takes place only
at a
thin fíame región
Z = Z, to be
deter-
mined
as a
function
of B. The
región
Z < Z
f
,
toward
the
fuel side,
is a
región
of
near-equi 1ibrium,
Y = 0,
while
the
chemical reaction
is
frozen
in the
región
Z
>
Z_, where
the
fuel
and
oxidizer coexist. Thus fuel leaks through
the
reaction
zone toward
the
oxidizer side.
The
mass fraction
of
oxidizer
on the
near-equi1ibrium
side
of the
fíame
is
small
of
order
T /T for Y /v of
order
1 but
becomes negligible
o
a o°°
3
if
Y /v «/.
Ooo
The mass fractions Y and Y are then given in terms of X = Z /y/2D by the relations
Y
Q
= 0, Y
p
= 1 - A
1
(2 - erfc X /l ) (6)
for X < X
f
= Z
f
/¡TIF , and
Y = Y [1 -
er
t
c
* ] , Y = A, erfc X /l (7)
o ooo erfc Xf F 2
for X > X
f
336
The continuity
of the
fuel concentration
at the
fíame implíes that
A
2
erfc
X /\T= 1 - A
}
(2
-
erfc
X Si) = Y
pf
(8)
The mass consumption rate
of
oxidizer
per
unit fíame surface
is
given
by:
, , exp (-xf)
(pD
Y ) = pD Y
A72rJ-£-
J-; (9)
z
Zf o» y—
erfc
X
r
Fuel
and
oxidizer
are
consumed
at
stoichiometric proportíons
at the
fíame,
so
that
exp{X^
(1 - L)}
A.
- A, = /L (Y /v) f—v (10)
1
2 'o»
erfc
X
f
and taking into account
Eq
(8)
we
obtain
erfc(X-
/L)
2A.
= 1 +
/T(Y^ /v)
a
„
f
/
Y
exp {X¿
(L-1)}
(11)
i
0°°
erre
A~ T
thus determiníng
the
concentration distribuíions
in
terms
of the
fíame position
X_
that will
be
found below
in
terms
of the
Damkohler number from
an
analysis
of the
reaction zone structure.
To calcúlate
the
temperature distribution
we
notice that
Y /v + c T/Q
behaves
as an
o p
inert species, so that
c (T-T )/Q + Y /v = (Y /2v)(2-erfc X) (12)
p o o o°°
In particular,
the
fíame temperature
T, is
given
by
T
f
- T
o
= (Q
Y
ooo
/vc
p
)
(1
- j
erfc
X
f
) (13)
Before giving
the
results
of the
analysis
of the
reaction zone structure,
let us
índicate that
the
outer structure
of the
diffusion fíame,
in the
d¡ffusion-contrelled
Burke-Schumann limiting case, B/y*
00
,
is
given
by the
above equations
if we
write
A„
= 0, to
insure that
no
fuel leaks toward
the
oxidizer side
of the
reaction zone.
Then,
the
fíame position
is X, = X ,
given
by the
relation
Y
™
erfc
x
„
ex
P
fxfo-i-)}
-°?L
S ! , (12.)
V
/T(2-erfc
X /T)
e
and
the
fíame temperature
T- = T
given
by
c
(T - T
)v/Q
Y = 1 - ¿
erfc
X (15)
p
e o o» 2 e
337
Only in the case L=1 does the fíame temperature in the diffusion controlled regime
coincide with the adiabatic fíame temperature T
ea
, given by:
c (T - T )v/0_ Y = (1 + Y
/v)"
1
,
(16)
p ea o o°° o°°
If Y /v is small the temperature gradient at the fíame toward the fuel side is
QCO
smáll compared with the gradient toward the oxidizer side. This simplifies the
anal-
ysisof the reaction zone structure in the premixed fíame regime given by Liñán
(197't) which can be used without changes in this case to give the following relation
— 1TV
Y
v c T_
P
t
>
0 Y T
2
2 2
Y
c
[erfc X,] exp(2X
¿
, - T /T,) = 1 (17)
re i t a t
between the Damkohler number and fíame position.
In summary the system of algebraic equations (8), (11), (13) and (17) determines
in parammetric form, with the fíame position X, as parameter, the relation between
the fíame temperature and the Damkohler number B/Y involving the straining rate;
this relation if, as it has been assumed, T,/T is large is multi-valued for valúes
of B/ >
Y(B/
) an extinction valué below which neither the premixed fíame regime
ñor the diffusion controlled regime is possible.
The effect of the Lewis number on the relation between the fíame temperature and
Damkohler number is shown in the figure for the promixed fíame regime. The lower
branch of each curve corresponds to unstable solutions of the equations.
The multiplicity of solutions would not exist if time derivative terms representing
transient effects were retained in Eqs.
0)-(3).
The time derivatives must
necessari'ly be retained if the straining rate y changes with time; however a
coordínate transformation as the one used by Coreos (1980) can be used to transform
the problem to that of the unsteady unstrained mixing layer between two half-spaces
of fuel and oxidizer, with a slight modification of the reaction rate term. Then the
analysis by Liñan and Crespo (1976) can be used to describe the mixing layer
structure.
Acknowledgement
This research has been partially supported by the U.S. Army European Office under
Grant DA-ERO-79 - G
0007.