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The concept of flame stretch
COMBUSTION A N D F L A M E 31,209-211 (1978)
209
BRIEF COIVIMUNICATIONS The Concept of Flame Stretch ROGER A. STREHLOW Aeronautical and A stronau tieal Engineering Department and
LESTER D. SAVAGE Mechanical & Industrial Engineering Department, University of lllinois at Urbana-Champaign, Urbana, Illinois
A laminar flame propagating in a premixed combustible mixture is a wave in the true sense of the word and, therefore, has the property that any element of the front propagates only normal to itself in any flow situation. However, flames are rather thick, and there are many flow situations where the equations for strictly one-dimensional flow through the flame are not applicable. Under these conditions the flame is said to exhibit stretch (either positive or negative). The first attempt to calculate the positive stretch factor for curved flames was made by Karlovitz et al [1]. Based on this work, Lewis and Von Elbe [2] derived formulas for the stretch factor in two geometries. In this paper we will redefine and rederive relationships for flame stretch. These analyses consider a hypothetical flame, which is infinitely thin but yet has a definable preheat zone thickness, r/o, and a normal burning velocity, S~, which are both invariant with curvature. Note that the assumption of an infinitely thin flame is incompatible with the assumption that the flame has a preheat zone. However, in these derivations, for simplicity, we ignore the perturbation of the flow near the flame introduced by the presence of the pereheat zone. As in the earlier derivations of stretch, the flame will be said to have positive stretch if its Copyright © 1978 by The Combustion Institute Published by Elsevier North-Holland, Inc.
frontal area increases with time due to flow geometry and negative stretch if it decreases with time. Under these conditions one can form the dimensionless group K.
a(ln AA) r/o . . . at S~
(1)
where ~(In A A ) / a t is a fractional rate of flame area increase with units of inverse seconds and r/o/Su is a characteristic time of propagation with units of seconds. Equation (1) represents a logical definition of the Karlovitz number, K, based on the behavior of an element of the wave front as it propagates in various approach situations. The Karlovitz number then indicates "stretch," being zero for no stretch. The value of the Karlovitz number for a number of specific simple flow situations will now be derived and compared to the existing derivations.
Stretch in Specific Flow Geometries I. The steady flame in an approach flow which is strictly two dimensional and contains a simple velocity gradient In this case, the approach flow is assumed to have a velocity vector U, which lies in the x direction
0010-2180/78/0031- 209501.25
R. A. STREHLOW and L. D. SAVAGE
210
~7 ÷ (÷) _
~ \\
1
t u x
~--
Fig. I. Vector diagram for a stretched oblique flame.
only and which contains a gradient dU/dY. The infinitely thin flame is assumed to have an orientation which makes it appear steady at every location (see Fig. 1). In this geometry, an element of the flame front is slipping along the flame front at the velocity UII = Su tan a. Since the geometry is two-dimensional the rate of area increase may be derived as follows: consider two points on the flame front which are separated by a distance As = s2 -- sl. The area of a unit of depth D of this flame section is AA = A~D and - D
at
-
-
at
=
(
(UII)o +
au,, as
A s - - (UII)
o)
D
Therefore, in this case
In
another
approach
one
can
substitute
Ull = Su tan a to obtain - - sec 2 o~, K =% R
(4)
where R is the local radius of curvature of the flame. The first term on the right hand side of Eq. (2) is the result obtained by Karlovitz et al., and presented in Lewis and Von Elbe. However, we obtained Eq. (3) which, though markedly different for small a, approaches the same limit value for large a. The reason for this difference is that the original derivation does not recognize that the flame angle a also changes as the oblique flame sheet slips along itself at the velocity Ull. In Reed's [3] application of the stretch concept to blowoff limit correlation, he replaces U by S u. This substitution actually changes Eq. (3) to aU~?o K = - - - - cot ~.
0 I n (AA)_ (aU,,
ay s .
a,
Thus, in the Reed work, a = 45 ° is the only condition at which
or
K-
Su \ as
0 u r/o
K = --,---sin a + - - - - c o t a cos a,
ayu
ayu
ay S. "
II. The growing spherical flame from spark ignition
Since UIi = U sin a, we obtain
~u~
aUto
(2)
The new definition of K (Eq. 1) yields an expression identical to that in Lewis and Von Elbe [2] for this geometry.
IlL Upward propagating flame cap in a cylindrical tube
or Ou 7 o
1
ay u sin a "
(3)
In this case, the flame is quasi-steady in coordinates that move with it, has cylindrical symmetry,
BRIEF COMMUNICATIONS
211 At, or near, the center line the flame cap may be assumed to have a constant radius of curvature, thus as a ~ 0 2r/o K+-R Thus, we have established that stretch exists at the center line of such a rising flame cap. In the other limit we can consider the skirt of the rising flame cap as having a limit shape which is a section of a cone. Under these circumstances R ~ oo in the direction of slip and
Fig. 2. Cylindrical geometry, flame propagating upward or outward. K
and is influenced both by the usual stretch due to its oblique orientation to the cold flow vector and by its cylindrical geometry (see Fig. 2). In this simple derivation, we assume that the approach flow vector U is always parallel to the center line of the tube, even though this is not really the case when a flame propagates in this geometry. Therefore ( ar - 27rr a_As)+ 2rrAs - at at at '
Since for this flame the cold approach flow is inside the cone the Karlovitz number becomes
K
ar - - = U cos or, at
0/.
REFERENCES
and
1.
a(zxs) au
B. Karlovitz, D. W. Denniston, D. H. Knapschaefer and F. E. Wells, Fourth Symposium (International)
on Combustion. Williams and Wilkins, Baltimore,
MD (1953), pp. 613-619.
as
2.
Thus, if the flame has a local radius of curvature R in the plane of the center line, we obtain (seR2a
1"/0 . = -- --Sln r
Since a negative value for stretch represents a "compression" of the flame, this flame is not stretched but is instead compressed.
however,
K=%
7/0 . -s i n or. r
IV. The conical bunsen burner flame on a MacheHebra nozzle not near the tip or homing region
aAA
at
+
sin a ) +-. r
(5)
3.
B. Lewis and G. Von Elbe, Combustion, Flames, and Explosions o f Gases, Academic Press, New York, NY
(1961), 2nd Ed., pp. 224-227,332-337. S.B. Reed, Combustion and Flame 11, 117-t89 (1967).
Received 20 April 1977; revised 21 July 1977
209
BRIEF COIVIMUNICATIONS The Concept of Flame Stretch ROGER A. STREHLOW Aeronautical and A stronau tieal Engineering Department and
LESTER D. SAVAGE Mechanical & Industrial Engineering Department, University of lllinois at Urbana-Champaign, Urbana, Illinois
A laminar flame propagating in a premixed combustible mixture is a wave in the true sense of the word and, therefore, has the property that any element of the front propagates only normal to itself in any flow situation. However, flames are rather thick, and there are many flow situations where the equations for strictly one-dimensional flow through the flame are not applicable. Under these conditions the flame is said to exhibit stretch (either positive or negative). The first attempt to calculate the positive stretch factor for curved flames was made by Karlovitz et al [1]. Based on this work, Lewis and Von Elbe [2] derived formulas for the stretch factor in two geometries. In this paper we will redefine and rederive relationships for flame stretch. These analyses consider a hypothetical flame, which is infinitely thin but yet has a definable preheat zone thickness, r/o, and a normal burning velocity, S~, which are both invariant with curvature. Note that the assumption of an infinitely thin flame is incompatible with the assumption that the flame has a preheat zone. However, in these derivations, for simplicity, we ignore the perturbation of the flow near the flame introduced by the presence of the pereheat zone. As in the earlier derivations of stretch, the flame will be said to have positive stretch if its Copyright © 1978 by The Combustion Institute Published by Elsevier North-Holland, Inc.
frontal area increases with time due to flow geometry and negative stretch if it decreases with time. Under these conditions one can form the dimensionless group K.
a(ln AA) r/o . . . at S~
(1)
where ~(In A A ) / a t is a fractional rate of flame area increase with units of inverse seconds and r/o/Su is a characteristic time of propagation with units of seconds. Equation (1) represents a logical definition of the Karlovitz number, K, based on the behavior of an element of the wave front as it propagates in various approach situations. The Karlovitz number then indicates "stretch," being zero for no stretch. The value of the Karlovitz number for a number of specific simple flow situations will now be derived and compared to the existing derivations.
Stretch in Specific Flow Geometries I. The steady flame in an approach flow which is strictly two dimensional and contains a simple velocity gradient In this case, the approach flow is assumed to have a velocity vector U, which lies in the x direction
0010-2180/78/0031- 209501.25
R. A. STREHLOW and L. D. SAVAGE
210
~7 ÷ (÷) _
~ \\
1
t u x
~--
Fig. I. Vector diagram for a stretched oblique flame.
only and which contains a gradient dU/dY. The infinitely thin flame is assumed to have an orientation which makes it appear steady at every location (see Fig. 1). In this geometry, an element of the flame front is slipping along the flame front at the velocity UII = Su tan a. Since the geometry is two-dimensional the rate of area increase may be derived as follows: consider two points on the flame front which are separated by a distance As = s2 -- sl. The area of a unit of depth D of this flame section is AA = A~D and - D
at
-
-
at
=
(
(UII)o +
au,, as
A s - - (UII)
o)
D
Therefore, in this case
In
another
approach
one
can
substitute
Ull = Su tan a to obtain - - sec 2 o~, K =% R
(4)
where R is the local radius of curvature of the flame. The first term on the right hand side of Eq. (2) is the result obtained by Karlovitz et al., and presented in Lewis and Von Elbe. However, we obtained Eq. (3) which, though markedly different for small a, approaches the same limit value for large a. The reason for this difference is that the original derivation does not recognize that the flame angle a also changes as the oblique flame sheet slips along itself at the velocity Ull. In Reed's [3] application of the stretch concept to blowoff limit correlation, he replaces U by S u. This substitution actually changes Eq. (3) to aU~?o K = - - - - cot ~.
0 I n (AA)_ (aU,,
ay s .
a,
Thus, in the Reed work, a = 45 ° is the only condition at which
or
K-
Su \ as
0 u r/o
K = --,---sin a + - - - - c o t a cos a,
ayu
ayu
ay S. "
II. The growing spherical flame from spark ignition
Since UIi = U sin a, we obtain
~u~
aUto
(2)
The new definition of K (Eq. 1) yields an expression identical to that in Lewis and Von Elbe [2] for this geometry.
IlL Upward propagating flame cap in a cylindrical tube
or Ou 7 o
1
ay u sin a "
(3)
In this case, the flame is quasi-steady in coordinates that move with it, has cylindrical symmetry,
BRIEF COMMUNICATIONS
211 At, or near, the center line the flame cap may be assumed to have a constant radius of curvature, thus as a ~ 0 2r/o K+-R Thus, we have established that stretch exists at the center line of such a rising flame cap. In the other limit we can consider the skirt of the rising flame cap as having a limit shape which is a section of a cone. Under these circumstances R ~ oo in the direction of slip and
Fig. 2. Cylindrical geometry, flame propagating upward or outward. K
and is influenced both by the usual stretch due to its oblique orientation to the cold flow vector and by its cylindrical geometry (see Fig. 2). In this simple derivation, we assume that the approach flow vector U is always parallel to the center line of the tube, even though this is not really the case when a flame propagates in this geometry. Therefore ( ar - 27rr a_As)+ 2rrAs - at at at '
Since for this flame the cold approach flow is inside the cone the Karlovitz number becomes
K
ar - - = U cos or, at
0/.
REFERENCES
and
1.
a(zxs) au
B. Karlovitz, D. W. Denniston, D. H. Knapschaefer and F. E. Wells, Fourth Symposium (International)
on Combustion. Williams and Wilkins, Baltimore,
MD (1953), pp. 613-619.
as
2.
Thus, if the flame has a local radius of curvature R in the plane of the center line, we obtain (seR2a
1"/0 . = -- --Sln r
Since a negative value for stretch represents a "compression" of the flame, this flame is not stretched but is instead compressed.
however,
K=%
7/0 . -s i n or. r
IV. The conical bunsen burner flame on a MacheHebra nozzle not near the tip or homing region
aAA
at
+
sin a ) +-. r
(5)
3.
B. Lewis and G. Von Elbe, Combustion, Flames, and Explosions o f Gases, Academic Press, New York, NY
(1961), 2nd Ed., pp. 224-227,332-337. S.B. Reed, Combustion and Flame 11, 117-t89 (1967).
Received 20 April 1977; revised 21 July 1977