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A note on thermal explosion theory
February 1969
L~T~RS TO Tm~ EDrroRs
97
for a stoichiometric mixture of this fuel was 2085°K, the experimental values indicate that combustion is more efficient in this case than for the aluminium slurries and is virtually complete at the first observation station. Observed flame species are those to be anticipated from the theoretical equilibrium composition calculations't. A 73 weight per cent suspension of boron in JP-4 was obtained from the Fuels and Lubricants Branch of the Aero Propulsion Laboratory at Wright-Patterson Air Force Base, Ohio. Particle size was reported to be in the range of 2 to5 ~tm. Combustion of the 73 per cent slurry was impossible with the burner described above, because of its high viscosity. A 50 per cent suspension was burned, but with difficulty. The studies reported here are for a 35 weight per cent suspension having a viscosity of approximately 10 cP. The flame (5 crn diameter and 28 cm length) had a green colouration arising from the BO2 bands 5 near 0.55 ~tm. Average particle residence time in the flame was approximately 200 ms. Spectral radiation emitted by the flame is shown in Figure 3. The principal radiating species are seen to be BO, BOz, HBO2, CO and/or CO2, and H20. The infra-red peaks at 2.7 and 4"9 ~tm are attributed 6 to HBO2. Flame temperatures were measured at three positions in the combustion zone using BO2 emission at 0.55 lam. These are to be compared with the theoretical adiabatic flame temperature of 2276°K calculated for a stoichiometric mixture. The ra~her large discrepancy between the observed and predicted temperatures may be attributed in part to the incomplete oxidation of the boron and to some extent to the errors inherent in the measurement. However, the flame species detected are consistent with the thermodynamic calculations. In closing, the following statements appear appropriate. Visible emission from flames containing aluminium and magnesium may be attributed primarily to continuum radiation from hot metal or metal oxide particles. However, strong band systems for both MgO and A10 produce substantial emission in the visible and/or ultra-violet. Most of the visible emission in boron flames results from the very strong BO2 bands near 0.55 pro. ]For the thin flames discussed above, continuum radiation is negligible in comparison with molecular' emission from carbon dioxide, and water in the infra-red. M. E. MORRISONand K. SCHELLER Chemistry Research Laboratory, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio 45433
(Received June 1968 ~revi~ed October 1968)
References i MORIUSON, M. E. and SCrIELLL~,K. 'Liquid fuel-air combustion II--Spectral brightness and emissivity of stoichiometric and rich (JP.4)-airflames', Western States Section, The Combustion Institute: LaJo[la, California (25 April 1967) z BABROV,H. J. J. opt. Soc. Amer. 51,171 (1961) 3 P~rSE, R. W. B. and GAYDON,A. G. The Identification of Molecular Spectra, Wiley: New York (1963) 4 GORDON,S. and ZELEZNXK,F. J. 'A general IBM 704 or 7090 computer program for computation of chemical equilibrium com~o:,itions, rocket performance, and Chapman-Jouguet Detonations', NASA Ta! D-1737 (October i 963) s KASKAN,W. E., MACKI~NSn/,J. D. and MILLIKAN,R. C. J. chem. Phys.34. 570 (196I) 6 KASKAN,W. E. and Mn.LrgArq, R. C. Eighth Symposium (International) on Combustion, p 262. William~ and Wilkins: Baltimore (1962)
A Note on Thermal Explosion Theory. THE Semenov ~ theory of thermal explosion considers a self-heating material to possess a spatially averaged temperature T while it is immersed in a constant temperature inert medium of temperature To. The heat lost by the material to the medium is taken as I ( T - To) where l is effectively a heat transfer coefficient although it also depends on the surface area of the interface between the reactive
98
Le~reas To"mE mrroas
VoL 13
medium and the heat b a t h If the heat release rate by the reaction is R(T, P) where P is the pressure in the case of reacting gases, then as is well known the critical condition is given by
R(T~P)=
[1]
(OR/Or)e= l
To);
I(T~-
i.e. the tangency condition where T¢is the critical temperature for the reactants. However, the ignition curve as normally plotted is represented by an equation of the form
e =f(To)
[2]
and equations 1 can be regarded as a parametric representation of the curve 2 using the parameter T~ The parametric form is very useful in cases where the explicit dependence of R on T and P is not so simple that the elimination of T can be carried out explicitly, i.e. R is not simply Ap" exp ( - E~ RT) but may be a function of terms of this type when concurrent chemical reactions are occurring. The first of equations I can be solved explicitly for To, i.e. To = T, . -
[3]
R(T~ P)/l
which on taking the total derivative with respect to To gives us d To { ct(~_~) --- -dT~ 1 p +
c1(~) ~T~}/ ! r
[4]
which on using the second of equations I can be reduced to
1 ~/', ~ dP
dTo =
- 7
[5]
or more simply
00 ~
~
T
which is as far as we can go towards obtaining the equivalent of the explicit relation 2 in the general case. Equation 6 gives us the slope of the experimentally measurable ignition limit curve as a function of the pressure dependence of the heat release rate. In general if (OR/OP)r > 0 it will have a negative :slope as usual. However, if we pass into a region of decreasing pressure dependence the ignition limitcurve will pass through a point of infinite slope before entering a region of positive slope, e.g. the type of ignition curve obtained in H2-Oe, CO-O,, CS2--O2, $iH4-Oe, PH3-O2 etc. In Figure 1
P
a~
I 0
Mr
P
A
>0 ! !
d._gp= 0 d7 o
J L
F~
FIGURE 1
To FIGURE 2
February 1969
LL~rERSTOTHEEDITORS
99
at T'o and T'~ (~R/t~P)T will be zero, and examination ofd2P/dT~ will in principle enable one to distinguish between these two points. However, in practice the expression obtained is rather complex. It is not suggested that these upper limits are thermal in nature, but it is simply interesting to note that ignition curves of this sort do not necessarily require a branching chain reaction. At the same time it is worth noting that low pressure lobes such as those observed with hydrocarbons, aldehydes etc. (Figure 2) can never be accounted for by a theory of this type since at the tip of the lobe it appears that dP/dTo = 0 (although a discontinuity in slope cannot be ruled out) and this would require an infmite value of (t~R/~P)T. In fact there are good reasons 2 to believe that dP/d TO is discontinuous at the points A and B in Figure 2, due to the fact that the critical condition along AB represents the disappearance of an oscillatory non-explosive state. • B F. GRAY School of Chemistry, University of Leeds
(Received August 1968; revised September 1968)
References t SEMENOV,N. N. Chemical Kinetics and Chain Reactions. Pergamon: Oxford (1953) 2 G ~ v , B. F. In preparation
Hydrodynamic Effects in the Flame Spreading, Ignitabflity and Steady Burning of Liquid Fuels IN determining flame spreading rates across combustible liquid fuels at temperatures below their flash point, it was found convenient, as others have done 1, to float films of the combustible liquid on water. In the particular experiments of concern here, styrafoam chips were floated on the surface of the combustible (generally kerosine) and an iron oxide powder at the fuel/water interface. After the flame was ignited, the styrafoam actually flowed in the direction of the flame propagation, whereas the iron oxide particles flowed somewhat slower toward the flame and in the direction opposite to that of flame propagation. A convective motion ahead of the flame appears to be present and to play an important part in determining the rate of flame propagation. To the best of the authors' knowledge, Roberts", in order to explain the difference between his experimental results and a flame spreading theory based upon conduction in the liquid, is the only previous investigator to suggest convection as an important effect. At this time it has not been established definitely what creates the convective motion or cell which seems to be present. It could arise from surface tension induced forces (the Marangoni effect) and/or buoyancy (gravity) forces. The fact that there is hydrodynamic flow in the igniting liquid and that viscous forces would compete with any surface tension induced or buoyancy forces, as given by the Marangoni and Grashof numbers respectively, led the senior author to believe liquid viscosity would play an important part in determining the flame-spreading rates. Consequently, kerosine, and some pure fuels, were thickened with Vistanex, a polyisobutylene of molecular weight approximately 200000, to a maximum of three per cent by weight. The strong effect of viscosity on the flame spreading rate is shown in Figures 1 and 2. These are results taken in the most simple of experiments 3. More accurate data will be obtained with more refined experimental techmques. Nevertheless, the strong viscosity effect is evident. Figure 2 reports results with experiments in which only kerosine of 1 cm depth (without water) was used. Whereas comparison of Figures 1 and 2 would indicate, as suggested by others 1' 2, that a difference in propagation rates is due to the heat sink offered by the water, the question now arises as to whether the fuel thickness also alters the convective motions. The thickening agent should not affect the vapour pressure of the fuel and, indeed, the measured flash points were not altered. The effect of such agents on surface tension is not known at this time.
L~T~RS TO Tm~ EDrroRs
97
for a stoichiometric mixture of this fuel was 2085°K, the experimental values indicate that combustion is more efficient in this case than for the aluminium slurries and is virtually complete at the first observation station. Observed flame species are those to be anticipated from the theoretical equilibrium composition calculations't. A 73 weight per cent suspension of boron in JP-4 was obtained from the Fuels and Lubricants Branch of the Aero Propulsion Laboratory at Wright-Patterson Air Force Base, Ohio. Particle size was reported to be in the range of 2 to5 ~tm. Combustion of the 73 per cent slurry was impossible with the burner described above, because of its high viscosity. A 50 per cent suspension was burned, but with difficulty. The studies reported here are for a 35 weight per cent suspension having a viscosity of approximately 10 cP. The flame (5 crn diameter and 28 cm length) had a green colouration arising from the BO2 bands 5 near 0.55 ~tm. Average particle residence time in the flame was approximately 200 ms. Spectral radiation emitted by the flame is shown in Figure 3. The principal radiating species are seen to be BO, BOz, HBO2, CO and/or CO2, and H20. The infra-red peaks at 2.7 and 4"9 ~tm are attributed 6 to HBO2. Flame temperatures were measured at three positions in the combustion zone using BO2 emission at 0.55 lam. These are to be compared with the theoretical adiabatic flame temperature of 2276°K calculated for a stoichiometric mixture. The ra~her large discrepancy between the observed and predicted temperatures may be attributed in part to the incomplete oxidation of the boron and to some extent to the errors inherent in the measurement. However, the flame species detected are consistent with the thermodynamic calculations. In closing, the following statements appear appropriate. Visible emission from flames containing aluminium and magnesium may be attributed primarily to continuum radiation from hot metal or metal oxide particles. However, strong band systems for both MgO and A10 produce substantial emission in the visible and/or ultra-violet. Most of the visible emission in boron flames results from the very strong BO2 bands near 0.55 pro. ]For the thin flames discussed above, continuum radiation is negligible in comparison with molecular' emission from carbon dioxide, and water in the infra-red. M. E. MORRISONand K. SCHELLER Chemistry Research Laboratory, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio 45433
(Received June 1968 ~revi~ed October 1968)
References i MORIUSON, M. E. and SCrIELLL~,K. 'Liquid fuel-air combustion II--Spectral brightness and emissivity of stoichiometric and rich (JP.4)-airflames', Western States Section, The Combustion Institute: LaJo[la, California (25 April 1967) z BABROV,H. J. J. opt. Soc. Amer. 51,171 (1961) 3 P~rSE, R. W. B. and GAYDON,A. G. The Identification of Molecular Spectra, Wiley: New York (1963) 4 GORDON,S. and ZELEZNXK,F. J. 'A general IBM 704 or 7090 computer program for computation of chemical equilibrium com~o:,itions, rocket performance, and Chapman-Jouguet Detonations', NASA Ta! D-1737 (October i 963) s KASKAN,W. E., MACKI~NSn/,J. D. and MILLIKAN,R. C. J. chem. Phys.34. 570 (196I) 6 KASKAN,W. E. and Mn.LrgArq, R. C. Eighth Symposium (International) on Combustion, p 262. William~ and Wilkins: Baltimore (1962)
A Note on Thermal Explosion Theory. THE Semenov ~ theory of thermal explosion considers a self-heating material to possess a spatially averaged temperature T while it is immersed in a constant temperature inert medium of temperature To. The heat lost by the material to the medium is taken as I ( T - To) where l is effectively a heat transfer coefficient although it also depends on the surface area of the interface between the reactive
98
Le~reas To"mE mrroas
VoL 13
medium and the heat b a t h If the heat release rate by the reaction is R(T, P) where P is the pressure in the case of reacting gases, then as is well known the critical condition is given by
R(T~P)=
[1]
(OR/Or)e= l
To);
I(T~-
i.e. the tangency condition where T¢is the critical temperature for the reactants. However, the ignition curve as normally plotted is represented by an equation of the form
e =f(To)
[2]
and equations 1 can be regarded as a parametric representation of the curve 2 using the parameter T~ The parametric form is very useful in cases where the explicit dependence of R on T and P is not so simple that the elimination of T can be carried out explicitly, i.e. R is not simply Ap" exp ( - E~ RT) but may be a function of terms of this type when concurrent chemical reactions are occurring. The first of equations I can be solved explicitly for To, i.e. To = T, . -
[3]
R(T~ P)/l
which on taking the total derivative with respect to To gives us d To { ct(~_~) --- -dT~ 1 p +
c1(~) ~T~}/ ! r
[4]
which on using the second of equations I can be reduced to
1 ~/', ~ dP
dTo =
- 7
[5]
or more simply
00 ~
~
T
which is as far as we can go towards obtaining the equivalent of the explicit relation 2 in the general case. Equation 6 gives us the slope of the experimentally measurable ignition limit curve as a function of the pressure dependence of the heat release rate. In general if (OR/OP)r > 0 it will have a negative :slope as usual. However, if we pass into a region of decreasing pressure dependence the ignition limitcurve will pass through a point of infinite slope before entering a region of positive slope, e.g. the type of ignition curve obtained in H2-Oe, CO-O,, CS2--O2, $iH4-Oe, PH3-O2 etc. In Figure 1
P
a~
I 0
Mr
P
A
>0 ! !
d._gp= 0 d7 o
J L
F~
FIGURE 1
To FIGURE 2
February 1969
LL~rERSTOTHEEDITORS
99
at T'o and T'~ (~R/t~P)T will be zero, and examination ofd2P/dT~ will in principle enable one to distinguish between these two points. However, in practice the expression obtained is rather complex. It is not suggested that these upper limits are thermal in nature, but it is simply interesting to note that ignition curves of this sort do not necessarily require a branching chain reaction. At the same time it is worth noting that low pressure lobes such as those observed with hydrocarbons, aldehydes etc. (Figure 2) can never be accounted for by a theory of this type since at the tip of the lobe it appears that dP/dTo = 0 (although a discontinuity in slope cannot be ruled out) and this would require an infmite value of (t~R/~P)T. In fact there are good reasons 2 to believe that dP/d TO is discontinuous at the points A and B in Figure 2, due to the fact that the critical condition along AB represents the disappearance of an oscillatory non-explosive state. • B F. GRAY School of Chemistry, University of Leeds
(Received August 1968; revised September 1968)
References t SEMENOV,N. N. Chemical Kinetics and Chain Reactions. Pergamon: Oxford (1953) 2 G ~ v , B. F. In preparation
Hydrodynamic Effects in the Flame Spreading, Ignitabflity and Steady Burning of Liquid Fuels IN determining flame spreading rates across combustible liquid fuels at temperatures below their flash point, it was found convenient, as others have done 1, to float films of the combustible liquid on water. In the particular experiments of concern here, styrafoam chips were floated on the surface of the combustible (generally kerosine) and an iron oxide powder at the fuel/water interface. After the flame was ignited, the styrafoam actually flowed in the direction of the flame propagation, whereas the iron oxide particles flowed somewhat slower toward the flame and in the direction opposite to that of flame propagation. A convective motion ahead of the flame appears to be present and to play an important part in determining the rate of flame propagation. To the best of the authors' knowledge, Roberts", in order to explain the difference between his experimental results and a flame spreading theory based upon conduction in the liquid, is the only previous investigator to suggest convection as an important effect. At this time it has not been established definitely what creates the convective motion or cell which seems to be present. It could arise from surface tension induced forces (the Marangoni effect) and/or buoyancy (gravity) forces. The fact that there is hydrodynamic flow in the igniting liquid and that viscous forces would compete with any surface tension induced or buoyancy forces, as given by the Marangoni and Grashof numbers respectively, led the senior author to believe liquid viscosity would play an important part in determining the flame-spreading rates. Consequently, kerosine, and some pure fuels, were thickened with Vistanex, a polyisobutylene of molecular weight approximately 200000, to a maximum of three per cent by weight. The strong effect of viscosity on the flame spreading rate is shown in Figures 1 and 2. These are results taken in the most simple of experiments 3. More accurate data will be obtained with more refined experimental techmques. Nevertheless, the strong viscosity effect is evident. Figure 2 reports results with experiments in which only kerosine of 1 cm depth (without water) was used. Whereas comparison of Figures 1 and 2 would indicate, as suggested by others 1' 2, that a difference in propagation rates is due to the heat sink offered by the water, the question now arises as to whether the fuel thickness also alters the convective motions. The thickening agent should not affect the vapour pressure of the fuel and, indeed, the measured flash points were not altered. The effect of such agents on surface tension is not known at this time.