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On the theory of laminar flames
Marclv 1964
Letters to the E d i t o r s
References J. a n d VAN
I VAN WONTERGHEM,
TIGGELEN, A. Bull.
Soc. cAirn. Belge, 1955, 64, 780 S. a n d SKACHKOV, G . I. Zh. prihl. Mekh. i tekh. Fiz. 1960, 2, 93 3 PANNETIER, G. and SICARD, A . F i l t h S y m p o s i u m (International) on Combustion, p 620. Reinhold:
2 ROZANCHENKO, L .
New York. 1955 4 FENIMORE, C. P. and JONES, G. W . 1959, 63, 1154 5 DIXON-LEwis, G. a n d ISLES, G. L. Soc. 1957, 53, 193 6 DIXON-LEWIS, G. a n d WILLIAMS, C~ Flame, 1960, 4, 383 7 GAYDON, A. G. Spectroscopy of C h a p m a n and Hall: L o n d o n , 1957
J. phys. Chem. Trans. Faraday A. Combustion
Flames,
p 87.
On the Theory of Laminar Flames IN A n u m b e r of papers by the author 1-'~ equations were obtained for flame velocities u 0 in systems with branched and unbranched chain reactions. The solutions were based on approximation of the known formula of A. N. KOLMOGOROV, I . G. PETROVSKII a n d I . S. PISKUNOV 6
87
heat capacity ( c a l / g degree), (I) the rate of heat release, T = T ' - To', T" the temperature (°K). The index 0 denotes the initial temperature of the starting mixture (T'=To'), while b is the burning temperature ( T ' = T d ) , and ,,~ is the temperature T,~'=0.5 ( T j + To' ) corresponding to the m a x i m u m temperature gradient. The range of possible application of equation 1 was determined in the work of ref. 3 and the approximate evaluation of its errors for various (f, functions was described. However, the relations ~-'~ for u 0 are used for the determination of experimental constants for elementary reactions, and thus it would be desirable to establish the exact values of errors given by equation 1 within a range of parameters and functions of practical importance. Equation 2 with conditions 3 and 4 was integrated by means of an electronic computer. The (I, function was taken as
to the relation B Uo=
q)
2 [A,,(I,,,/' ,"-' - =
Copol r,,, ]
....
.
(or the lower value of the set of eigenvalues in the energy conservation equation
dT"
d(dT')
~dx X-(ix
-BC~d.~ +`1'=0
. . . [2J
for the conditions :
x=-oo, x--oo,
dT" dx = 0
T'=T,,',
.. [3]
dT'
T'=T,,',
dx =0
3 000 2 500 2 000 1 500
[s]
.
Here h is the heat of reaction (cal/mole), is the initial substance concentration ( m o l e / g mixture), k(T')=Aexp(-E/RT') is the rate constant (era :~/mole see), nb is the concentration of active centres at T ' - T b " (mole/g mixture). The following values were taken for H = hnb/ {/~ ( T ( - T.')} H - I . 1 6 × 10 -~ at H = 1 . 5 8 × 1 0 -'~ at H = l . 0 4 × 1 0 -'~ at H = 1 . 3 6 × 1 0 -7 at In
integrating
it was
T~'-3000°K T~,':2500°K T~,'=2000°K
T(=IS00°K
always
J \ o = 10--4
assumed
numerical
from relation
integration
1
226 68'5 13"7 1"03
202 66'7 14"1 1"12
from from eq. use of 2 relation I numerical 1 lintegration per cent --1 +3 -{-9
A =2×
that
10 - 4 error
error
error
from eq.
.
Burning velocities Uo (cm /see) at A = I0 la and E = l0 O00
Ao=SX 10-5
T" ° K
.
n t=(1/,l~ ) ( T ( - T ' ) / ( T ( - T o " )
. . ~4]
the definitions being: B = u p = u , , p 0 , u the velocity (cm/sec), p the density (g/cm'~), X the heat conductivity (cal/cm sec degree), c the Table 1.
hf,'-'k (T') n~ (T') nb =HI' 2 (Tb'-- T') k (T')
/
98 18"6 1 "37
from rela;ion
from use of relation I
from eq. 2 numerical integration
from from relation I
--11 --4 +7 +16
I
her cent
per cent 285 94-5 19"9 1 "59
use Of relation
447 137 26.2 1.94
402 134 28-1 2-25
-lO --2 +7 +16
Letters to the E d i t o r s
88
Vol.
Table 2. Burning velocities uo (cm/sec) at A = 2 × 1014 and E = 2 0 000 A = 5 × 10 -5
A.o = 10-~ error
• T" ° K
b
3 2 2 1
000 500 000 500
from eq. 2 numerical integration 201
from relation 1
from
7"78 0'341
2
from
relation numerical] relation 1 per lent integration:
196 49.4 7"16 0'307
51-7
from eq.
use o/
~--2 --4 -8
-10
I--285-/ 73'2 / 11"0
I
275 69"8 10'1 0"435
0.457
h 0 = 2 X 10-* error from use of
relation l
per cent -4
-5 -8 -5
from eq. 2 numerical integration
from
use of
relation
relation
1
Jper cent 403 103 15"4 0"739
391 95"3 14-1 0"618
-3 --7 - 8 - 16
Table 3. Burning velocities uo (cm/sec) at A=IO a:' and E = 2 0 000 h 0 = S X 10 -5
h0=10-1 I
error 1",b o K
3 000 2 500 2 000 1 500
from eq. 2 numerical integration 452 116 17"5 0"798
from
from
relation I
438 111 15"8
0"694
from eq. [ 2 t from relaltion numerical I relation integration[I 1 cent Oer --3 637 622 --5 , 163 157 use Of
--10 --13
24"6 1"02
k ( T ' ) = A [exp ( - E / R T ' ) - e x p (-E/RT[,')], in order that at T ' - T . ' , ([, (To") shall always be zero. The other parameters and functions are: h=10% c=0"3, R = 1.987, #=24, p = # / 8 2 . 0 6 T', X=)% (T'/To')°'% The burning velocity values calculated by means of an electronic computer from equation 2 and those obtained from relation 1 are listed in Tables I to 3. Analysis of the data in Tables I to 3 shows that the approximate relation 1 permits determination of the burning velocity with fair accuracy. T h e a u t h o r wishes S. I. K h u d y a e v and 9f the given p r o b l e m c o m p u t e r , and to V. &scussions.
to express his gratitude to Z. B . Maiofis /or solution b y m e a n s of an electronic Y a . B a s e v i c h for helpful L . A . LOVACHEV
22"3 0-979
A a = 2 X 10-'l error
from use Of
relation 1
per cent --2 --4 --9
--2
error
/rom eq. 2 numerical integration 902 231 34"4 1"65
from
relation l 875 221 31"5 1 '39
from use of
relation 1 per cent --3 --4 --8 -- 16
Institute of Chemical Physics, Academy of Sciences, Vorobyevskoye chaussde 2-b, Moscow V-334, U.S.S.R. (Received December 1963) References J LOVACHEV, L. A. Dokl. Ahad. Nauk S.S.S.R.. 1958, 120, 1287 LOVACHr:V, L. A. Izvest. Ahad. Nauk S.S.S,R., Otdel. khim. Nauk No. 10, 1959~ 1740 :x LOVACHEV, L. A. Combustion 6~ Flame, 1960, 4, 357 t LOVACrtEV, L. A. Eighth Symposium (International) on Combustion, p p 411-417. Williams a n d Wilkins: Baltimore, 1962 5 LOVACrIEV, L. A. Eighth Symposium (International) on Combustion, p p 418-426. Williams a n d Wilkins: Baltimore, 1962 6 KOLMOGOROV, A. N . , PETROVSKII, I . G . a n d PISKUNOV, I. S. Bull. Mosk. Gos. Universit., sect. A, 1937, 1~ No. 6
Letters to the E d i t o r s
References J. a n d VAN
I VAN WONTERGHEM,
TIGGELEN, A. Bull.
Soc. cAirn. Belge, 1955, 64, 780 S. a n d SKACHKOV, G . I. Zh. prihl. Mekh. i tekh. Fiz. 1960, 2, 93 3 PANNETIER, G. and SICARD, A . F i l t h S y m p o s i u m (International) on Combustion, p 620. Reinhold:
2 ROZANCHENKO, L .
New York. 1955 4 FENIMORE, C. P. and JONES, G. W . 1959, 63, 1154 5 DIXON-LEwis, G. a n d ISLES, G. L. Soc. 1957, 53, 193 6 DIXON-LEWIS, G. a n d WILLIAMS, C~ Flame, 1960, 4, 383 7 GAYDON, A. G. Spectroscopy of C h a p m a n and Hall: L o n d o n , 1957
J. phys. Chem. Trans. Faraday A. Combustion
Flames,
p 87.
On the Theory of Laminar Flames IN A n u m b e r of papers by the author 1-'~ equations were obtained for flame velocities u 0 in systems with branched and unbranched chain reactions. The solutions were based on approximation of the known formula of A. N. KOLMOGOROV, I . G. PETROVSKII a n d I . S. PISKUNOV 6
87
heat capacity ( c a l / g degree), (I) the rate of heat release, T = T ' - To', T" the temperature (°K). The index 0 denotes the initial temperature of the starting mixture (T'=To'), while b is the burning temperature ( T ' = T d ) , and ,,~ is the temperature T,~'=0.5 ( T j + To' ) corresponding to the m a x i m u m temperature gradient. The range of possible application of equation 1 was determined in the work of ref. 3 and the approximate evaluation of its errors for various (f, functions was described. However, the relations ~-'~ for u 0 are used for the determination of experimental constants for elementary reactions, and thus it would be desirable to establish the exact values of errors given by equation 1 within a range of parameters and functions of practical importance. Equation 2 with conditions 3 and 4 was integrated by means of an electronic computer. The (I, function was taken as
to the relation B Uo=
q)
2 [A,,(I,,,/' ,"-' - =
Copol r,,, ]
....
.
(or the lower value of the set of eigenvalues in the energy conservation equation
dT"
d(dT')
~dx X-(ix
-BC~d.~ +`1'=0
. . . [2J
for the conditions :
x=-oo, x--oo,
dT" dx = 0
T'=T,,',
.. [3]
dT'
T'=T,,',
dx =0
3 000 2 500 2 000 1 500
[s]
.
Here h is the heat of reaction (cal/mole), is the initial substance concentration ( m o l e / g mixture), k(T')=Aexp(-E/RT') is the rate constant (era :~/mole see), nb is the concentration of active centres at T ' - T b " (mole/g mixture). The following values were taken for H = hnb/ {/~ ( T ( - T.')} H - I . 1 6 × 10 -~ at H = 1 . 5 8 × 1 0 -'~ at H = l . 0 4 × 1 0 -'~ at H = 1 . 3 6 × 1 0 -7 at In
integrating
it was
T~'-3000°K T~,':2500°K T~,'=2000°K
T(=IS00°K
always
J \ o = 10--4
assumed
numerical
from relation
integration
1
226 68'5 13"7 1"03
202 66'7 14"1 1"12
from from eq. use of 2 relation I numerical 1 lintegration per cent --1 +3 -{-9
A =2×
that
10 - 4 error
error
error
from eq.
.
Burning velocities Uo (cm /see) at A = I0 la and E = l0 O00
Ao=SX 10-5
T" ° K
.
n t=(1/,l~ ) ( T ( - T ' ) / ( T ( - T o " )
. . ~4]
the definitions being: B = u p = u , , p 0 , u the velocity (cm/sec), p the density (g/cm'~), X the heat conductivity (cal/cm sec degree), c the Table 1.
hf,'-'k (T') n~ (T') nb =HI' 2 (Tb'-- T') k (T')
/
98 18"6 1 "37
from rela;ion
from use of relation I
from eq. 2 numerical integration
from from relation I
--11 --4 +7 +16
I
her cent
per cent 285 94-5 19"9 1 "59
use Of relation
447 137 26.2 1.94
402 134 28-1 2-25
-lO --2 +7 +16
Letters to the E d i t o r s
88
Vol.
Table 2. Burning velocities uo (cm/sec) at A = 2 × 1014 and E = 2 0 000 A = 5 × 10 -5
A.o = 10-~ error
• T" ° K
b
3 2 2 1
000 500 000 500
from eq. 2 numerical integration 201
from relation 1
from
7"78 0'341
2
from
relation numerical] relation 1 per lent integration:
196 49.4 7"16 0'307
51-7
from eq.
use o/
~--2 --4 -8
-10
I--285-/ 73'2 / 11"0
I
275 69"8 10'1 0"435
0.457
h 0 = 2 X 10-* error from use of
relation l
per cent -4
-5 -8 -5
from eq. 2 numerical integration
from
use of
relation
relation
1
Jper cent 403 103 15"4 0"739
391 95"3 14-1 0"618
-3 --7 - 8 - 16
Table 3. Burning velocities uo (cm/sec) at A=IO a:' and E = 2 0 000 h 0 = S X 10 -5
h0=10-1 I
error 1",b o K
3 000 2 500 2 000 1 500
from eq. 2 numerical integration 452 116 17"5 0"798
from
from
relation I
438 111 15"8
0"694
from eq. [ 2 t from relaltion numerical I relation integration[I 1 cent Oer --3 637 622 --5 , 163 157 use Of
--10 --13
24"6 1"02
k ( T ' ) = A [exp ( - E / R T ' ) - e x p (-E/RT[,')], in order that at T ' - T . ' , ([, (To") shall always be zero. The other parameters and functions are: h=10% c=0"3, R = 1.987, #=24, p = # / 8 2 . 0 6 T', X=)% (T'/To')°'% The burning velocity values calculated by means of an electronic computer from equation 2 and those obtained from relation 1 are listed in Tables I to 3. Analysis of the data in Tables I to 3 shows that the approximate relation 1 permits determination of the burning velocity with fair accuracy. T h e a u t h o r wishes S. I. K h u d y a e v and 9f the given p r o b l e m c o m p u t e r , and to V. &scussions.
to express his gratitude to Z. B . Maiofis /or solution b y m e a n s of an electronic Y a . B a s e v i c h for helpful L . A . LOVACHEV
22"3 0-979
A a = 2 X 10-'l error
from use Of
relation 1
per cent --2 --4 --9
--2
error
/rom eq. 2 numerical integration 902 231 34"4 1"65
from
relation l 875 221 31"5 1 '39
from use of
relation 1 per cent --3 --4 --8 -- 16
Institute of Chemical Physics, Academy of Sciences, Vorobyevskoye chaussde 2-b, Moscow V-334, U.S.S.R. (Received December 1963) References J LOVACHEV, L. A. Dokl. Ahad. Nauk S.S.S.R.. 1958, 120, 1287 LOVACHr:V, L. A. Izvest. Ahad. Nauk S.S.S,R., Otdel. khim. Nauk No. 10, 1959~ 1740 :x LOVACHEV, L. A. Combustion 6~ Flame, 1960, 4, 357 t LOVACrtEV, L. A. Eighth Symposium (International) on Combustion, p p 411-417. Williams a n d Wilkins: Baltimore, 1962 5 LOVACrIEV, L. A. Eighth Symposium (International) on Combustion, p p 418-426. Williams a n d Wilkins: Baltimore, 1962 6 KOLMOGOROV, A. N . , PETROVSKII, I . G . a n d PISKUNOV, I. S. Bull. Mosk. Gos. Universit., sect. A, 1937, 1~ No. 6