3395
SLOWOXIDATION OF HYDROCARBOX AND COOLFLAMES
On the Slow Oxidation of Hydrocarbon and Cool Flames
by C. H. Yang State University of New York at Stony Brook, Stony Brook, New York
11790
and B. F. Gray The Unisersity of Leeds, Leeds, England
(Received December $0,1968)
.A mathematical model is devised to consider the thermokinetic coupling of an active intermediate and temperature in the slow oxidation of hydrocarbon. With a single set of assumed rate constants it is shown that explosion limits, cool flame oscillation limits, oscillation period and amplitude, explosion peninsula (lobe), and negative temperature coefficient effect can all be calculated in agreement with experimentalobservations. Detail kinetics are discussed. The RCO radical is tentatively identified as the autocatalytic carrier. The proposed branching scheme conforms with a previous proposal made by Knox and Norrish in their work on the oxidation of propane.
I. Introduction The slow oxidation of hydrocarbon has been subjected to extensive experimental studies. The results of Newitt and Thornes,' Day and Pease,2 Knox and N ~ r r i s h , Bardwell ~,~ and Hinshelwood,j Sokolik and Yantovskiij6 etc., suggest that the reaction proceeds by two distinct mechanisms depending on temperature. The high-temperature region is about 400" and the lowtemperature region extends from 250 to 400". The transition between these two regions is often associated with interesting phenomena such as negative temperature coefficients, cool flames, oscillatory cool flames, low pressure, explosion peninsula in the pressure-bath temperature diagram, two-stage ignition, etc. All these diff erent characteristics are observed for the oxygen-hydrocarbon system of fixed composition in a rather narrow bath temperature range of about 100" and a pressure range from 200 or 300 mm to atmospheric pressures. One would expect that all these characteristics are interrelated and that the kinetics and energetics of the system would explain all the observations. Current theories of cool flames, twostage ignition, and negative temperature coefficients are fragmentary. They do not provide a general mathematical model or conceptual framework into which the simultaneous occurrence of all these unusual phenomena can be fitted. Keither the thermal nor the chain theory of explosion provides explanations to the two-stage ignition phenomenon and the unusual lowpressure explosion peninsula in the pressure-bath temperature diagram. There are two mathematical theories formulated to explain the periodicity of the cool flames. A kinetic oscillation theory was proposed by Frank-Kamenetski.' He has shown that under isothermal conditions, using a kinetic scheme that gives equations similar to those of the well-known animal population problem of Lotkajs a reacting system is capable of sustaining oscillations
between the concentrations of two intermediates. Frank-Kamenetski suggested that the oscillating intermediates are peroxides and aldehydes. Later attempts to identify these intermediates stimulated interesting discussion among Lewis and von Elbe,g Walsh,'" Bardwell and Hinshel~ood,~ and Knox and N o r r i ~ h . ~Their , ~ proposed kinetic schemes, however, have not been generally accepted. One important difficulty in accepting this theory for a gaseous kinetic system is the fact that the necessary condition of oscillation deduced from the theory is independent of temperature and pressure which is obviously a t variance with the experiment. The theory may be applicable to oscillations observed in liquid reactions where pressure and temperature changes may not be important. Salnikoff ,11 in contrast, proposed a thermokinetic theory which considers the oscillation between the temperature of the reacting medium and the concentration of a critical intermediate. With the very simple scheme
A--%X--%B
(1) D. &.I.Newitt and L. S. Thornes, J . Chem. Soc., 1656 (1937). (2) R. A. Day and R. N. Pease, J . Amer. Chem. Soc., 62, 2234 (1940). (3) J. H. Knox and R. G. W. Norrish, Trans. Faraday Soc., 50, 928 (1954). (4) J. H. KnoxandR. G. W. Norrish, Proc. Roy.Soc., A221,151 (1954). (5) J. Bardwell and C. N. Hinshelwood, ibid., A205, 375 (1951). (6) A. S. Sokolik and S. A. Yantovskii, Zh. Fiz. Khim., 20, 13 (1946). (7) D. A. Frank-Kamenetski, Dokl. Akad. Nauk SSSR, 25, 672 (1939). (8) A. J. Lotka, J.Amer. Chem. Soc., 42,1595 (1920). (9) B. Lewis and G. van Elbe, "Combustion Flames and Explosion of Gases," Academic Press, New York, N. Y., 1981. (10) A. D. Walsh, Trans. Faraday Soc., 42,264 (1946). (11) I. E. Salnikoff, Zh. Fiz. Khim.. 23,258 (1949).
Volume 78, Number 10 October 1969
3396 he was able to deduce the necessary condition of thermokinetic oscillation for this system. The condition is that the activation energies El and E2 of IC1 and k~ must satisfy the inequality E2 > El. Knox and Norr i ~ h subsequently ~,~ suggested that the critical intermediate may be the higher aldehydes, which was originally proposed by Newitt and Thornes.' The Salnikoff theory seems more consistent with the experimental observations than the theory of Frank-Kamenetski which assumed the oscillations to occur isothermally. The fact that oscillations coincide with significant temperature fluctuations indicates the thermal aspect is by no means negligible. The simple model devised by Salnikoff, however, failed to provide a formulism so that quantitative calculation can be made. It yields no additional insight to explain other characteristics such as negative temperature coefficient, explosion peninsula, etc. The present work followed the approach of Sanikoff in that it considered the thermokinetic coupling of a reactive intermediate and the temperature. The mathematical model is closely related to the one previously devised for unifying'2,l3 the theories of chain and thermal explosion. I n the present work, explosion, oscillation, and negative temperature coefficient effects are treated with a single kinetic scheme and the same system of thermokinetic equations. It will be shown that explosion and oscillation limits in PO-TO plane, oscillation frequency, induction periods, and negative temperature effects can be quantitatively calculated. The mathematical model is discussed in section 11. The oxidation process of hydrocarbon is assumed to be dominated by an active intermediate. A general kinetic scheme of production and consumption of the intermediate is postulated without the details of the elementary steps of reactions. The basic time-dependent kinetic and energy equations are derived. The thermokinetic steady states of these equations are shown to prescribe the various modes of oxidation, slow reaction, stable reaction, fast reactions with oscillations, unstable reaction (explosion), negative temperature coefficient effect, etc. To show that the mathematical model can make precise predictions of the oxidation characteristics, explosion limits, oscillation limits, period and amplitude of oscillations, and negative temperature coefficients are calculated with assumed rate constants in section 111. Results show that calculated values closely match the experimental observations.
11. Thermokinetic Model for the Slow Oxidation of Hydrocarbon Kinetic studies of the slow oxidation of hydrocarbons show that many species of intermediates and final products are produced during the reaction process. Many reactions are occurring concurrently during the oxidation process.14~16 It is widely accepted that a branchThe Journal of Physical Chemistry
C. H. YANGAND B. F. GRAY ing chain process4 is involved. The chain initiation, branching, and termination processes must be combinations of many of these concurrent reactions. It is also expected that some of these reactions may not participate in the chain process. I n this section we propose a mathematical model which considers the energy and the kinetic equations of a chain reaction. The scheme for initiation, branching, and termination of the chain is postulated as ki
Ah;X
initiation
x k bhb ' 2 x
branching
ktr.
x ht; stable products x
2
(1)
1
termination
stable products
The termination reaction is assumed to go through two parallel paths concurrently. With this kinetic scheme it will be shown that the explosion, oscillation, and negative temperature coefficient characteristics can be deduced from the mathematical model consisting of the energy and kinetic equations. Clearly, it is assumed in the present theory that the oxidation of a hydrocarbon is dominated by the autocatalytic carrier which we shall attempt to identify in later discussions. A . Therrnokinetic Steady States of Hydrocarbon Oxidation. Following kinetic scheme 1, the kinetic and energy equations become (with the concentration of primary reactants regarded constant for the early part of the reaction)
where
x = concentration of the intermediate, mol cc-l 6, = specific heat of the mixture under constant volume conditions, cal cc-1 OK-' = heats of reaction, cal mol-'
L
rh
= conductive heat loss = - (T
d2
- To),cal CC"
sec-l (12) B. F. Gray and C. H. Yang, J . Phys. Chem., 69,2747 (1965). (13) C. H.Yang and B. F. Gray, 11th International Symposium on Combustion, The Combustion Institute, Pittsburgh, Pa., 1967, p 1099. (14) J. H.Knox, Combust. Flame, 9,297 (1966). (15) N. N. Semenov, "Some Problems in Chemical Kinetics and Reactivity," Vol. I and 11, Princeton University Press, Princeton, N. J., 1959.
SLOWOXIDATION OF HYDROCARBON AND COOLFLAMES T
=
average temperature of the mixture in the vessel,
-
"K
To bath temperature, "K l? = heat exchange coefficient X = conductivity of the mixture, cal cm-' OK-' sec-l d = diameter of the vessel, cm Equations 2 and 3 are the basic equations for the system. The time history of the dependent variable temperature T and carrier concentration x can be deduced from eq 2 and 3 when both of the parameters, pressure Po and bath temperature To, are given. We assume that the rate constants and other physical constants can be independently estimated, of course. Since both eq 2 and 3 do not involve the time variable explicitly, solutions can most conveniently be studied in the T-x (phase) plane. Previous workers on cool flames16 and explosion^^^^^^ have made extensive use of this mathematical technique. The thermokinetic steady states of the system correspond to the singularities in the phase plane. The nature of the singularities in the phase plane furnishes much important information about the solutions in the neighborhood of the thermokinetic steady states. The singularities can be located by putting the time derivatives of eq 2 and 3 equal to zero
3397 The singularity xs,T , is a stable nodal point if the following inequalities are satisfied ( A , B , C , and D are evaluated a t the singularity).
B+C 0
(9)
I n the neighborhood of an unstable focus the system will diverge from it with oscillations when
+
ki(T8)hi
+
[B Cl > 0 [(B C)' 4(AD
+
+
- BC)]< 0
(10)
The thermokinetic steady states characterized by these singularities will now be used for determining the negative temperature coefficient, cool flame limits, and ignition limits in the pressure-bath temperature diagram. B. Self-Excited Thermokinetic Oscillation. Oscillations associated with the thermokinetic steady states occur in the neighborhood of focuses and vortex points. However, a self-excited oscillation occurs in a system which is not related to any of the thermokinetic states. The existence of such self-excited oscillations is established if the existence of limit cycles in the phase plane is shown.2 Limit cycles are either themselves closed solutions of eq 2 and 3 in the phase space x and T or they are the asymptotic limits of such solutions. I n the present study the existence of self-excited oscillation will be established through numerical integration of eq 2 and 3. C. The Negative Temperature Coeficient. The term negative temperature coefficient is used to indicate that the reaction rate of a system decreases with increasing temperature. Naturally, the temperature should be that of the reacting mixture. However, experimental measurements of the maximum rate of pressure rise dP/dt are typically taken to represent the maximum (16) I. E. Salnikoff, Zh. Fiz. Khim., 23,258 (1948). (17) H. T. Davis, "Introduction to Nonlinear Differential and Integral Equations," Dover Publications, New York, N. Y., 1962.
Volume 79,Number 10 October 1969
C. H. YANQ AND B. F. GRAY
3398 reaction rate. The bath temperature is generally taken to characterize that of the reacting mixture (isothermal assumption). It should also be noted the (dP/dt),,, can be regarded as the maximum reaction rate of the system only if heat losses to the bath can be neglected. When losses are considered, (dP/dt)m,x represents the maximum net energy release rate for a specific experiment with the heat loss rate of the system subtracted from the gross heat release rate. The maximum rate of the system is reached at the thermokinetic steady state. The positive terms in eq 3 represent the gross heat release R of the system. At the thermal steady state dT/dt = 0; therefore, R = L. dP/dt is also zero if the mole change in the reaction is neglected. The average temperature of the reacting mixture attains its maximum value at the thermokinetic steady state if the singularity is a stable nodal point. Consequently, the gross reaction rate also attains its maximum value a t this point. Since R = L a t the steady state which is proportional to (T, - TO),the plot of AT = T, - TO against To will represent the relationship between R and To. Knox and Norrisha plotted AT against TO as an alternative to (dP/dt),,, against Toin their work. The AT plot coincided with the dP/dt plot closely. For the present kinetic scheme a negative temperature coefficient is possible only if the condition Et1
< E b < Et2
(11)
is satisfied. The existence of the negative temperature coefficient can be qualitatively seen as follows. Let us assume Etl = 0 and xS is appreciable so that the term kihi can be neglected. The total heat release a t the thermokinetic steady state becomes
At low temperatures where the denominator is dominated by ktl, the rate is an increasing function of T because of the numerator. At high temperatures the denominator is dominated by the term kt2, and R is again an increasing function of T , provided Ei Eb > Et2. However, the denominator will pass through a minimum between the low- and high-temperature ranges if k b is greater than kt2 a t low temperatures. The minimum value of the denominator is attained at the temperature at which E b k b = Et2kt2. Therefore, R may pass through a maximum that corresponds to the minimum of the denominator, that is to say that the gross reaction rate R has a negative temperature coefficient. Before R becomes an increasing function with temperature again a t still higher temperatures, it must pass through a minimum. D. Coo1 Flame Oscillation and Oscillation Limits. A t a narrow range of bath temperature and pressure, the hydrocarbon and oxygen system is observed to develop oscillatory characteristics both in some in-
termediate concentrations as well as the pressure and temperature inside the reacting vessel. Associated with these oscillations there are multiple passages of cool flames. Is the critical compound which leads to the cool flame the same critical intermediate which causes the oscillation? This question has never been settled. Repa and Shtern1*,19hold the view that cool flame is a negligible side process. Lucquin20 and Cherneskey and BardwellZ1 also observed that cool flames in their study of the butane system do not always appear simultaneously with oscillations. A mechanism of cool flame is proposed in the present work. Generally we regard the cool flame as a side process except in the methane system where it is directly involved with the critical branching carrier. This will be discussed in more detail later. The oscillations between the concentration of the chain carrier and the temperature of the reacting gases will now be considered. The thermokinetic system prescribed by eq 2 and 3 will be oscillatory in the neighborhood of a thermokinetic steady state which is characterized by a focus singularity. The conditions for oscillations can be obtained in terms of bath temperature and initial pressure by studying the singularities (z,,T,) of the system in the (T,x)phase plane. Let us treat the case when the singularity is a stable focus. From previous results the two conditions of eq 10 must be satisfied for a stable focus. For the present kinetic scheme, condition 10 yields
bL bT
+
kb
- kti
+
kt2
+
The Journal of Physical Chemistry
Condition 13 guarantees characteristic roots having negative real parts; thus the system is stable. Condition 14 prescribes that the characteristic roots are complex, the necessary conditions for oscillatory trajectories. Equation 14 relates the two key parameters initial pressure Po and bath temperature To when the values of X, and T , are substituted. X, and T, are steady-state solutions of eq 5 and 6 in terms of (18) L.A, Repa and V. Ya. Shtern, Dokl. Akad. Nauk SSSR, 91,309 (1963). (19) L.A. Repa and V. Ya. Shtern, Zh. Fiz. Khim., 28, 415 (1954). (20) M.Lucquin, Compt. Rend., 250,854 (1960). (21) M.Cherneskey and J. Bardwell, Can. J . Chem., 38, 482 (1960).
SLOWOXIDATION OF HYDROCARBON AKD COOLFLAMES
TOand Po. Let us consider the curve represented by the equation [ ( B - C)'
+ 4AD] = 0
(15)
in the Po-To plane which, if existent, must divide the Po-To plane into two different regions. I n one of these regions for any values of Po and Tothe inequality
[ ( B - C)'
+ 4AD] > 0
(16)
is satisfied. In the other region the opposite is true
[ ( B - C)'
+ 4AD] < 0
I n the region where the above inequality which is the same as inequality 14 holds, the complex characteristic roots prescribe oscillatory trajectories for the intermediate concentration. I n other words, oscillatory cool flames should be observed. The curve of eq 15 in the Po-To plane, therefore, should be identified as the oscillation limits similar to the explosion limits. The regions it enclosed should contain all of the initial pressures and bath temperatures where two or more flames are observed. To satisfy the inequality 14 it is obvious that the term 4AD must be negative. This occurs only if the term bki/bT at x = xs and T = T , is very small in comparison to the rest of the derivatives of the rate constants. This is usually true when Po and To are significantly below the explosion limits. I n order that bktz/dT or bk&T be greater than bkb/dT one of the activation energies of these two termination reactions must be greater than E b , the activation energy of the branching reaction. This is also one of the necessary conditions for the existence of negative temperature coefficients discussed previously, and it is clear that the oscillations are always observed in the negative temperature coefficient zone in agreement with experiment. It can be easily deduced that the limit of oscillation prescribed by eq 15 is a closed curve in the Po-To plane. The region is bounded on the To axis within the zone of negative temperature coefficient. It is clear that the inequality 14 is not satisfied outside of the negative temperature coefficient region. On the Po axis the region is bounded on the top by the explosion limits. At very low pressures when all the reaction rates are small, inequality 16 is satisfied because bL/bT is insensitive to pressure; there is thus a lower bound in pressure below which no oscillation can occur. The oscillation frequency, w, can be easily estimated from the characteristic roots w
[ ( B - C)'
- 4AD]"'/2
(17)
The value of the damping factor for the oscillation can also be estimated as follows cc = ( B
+ C)/2
3399 will generally be observed experimentally as if it were a nodal singularity. Let the initial amplitude be I o and assume the oscillation can be approximated by the linear case as
I = IOeCteiW' (19) The period of the oscillation is 2 n / w . Let the damping per cycle, Q, be defined as 0 =
e-2r*/w
The initial amplitude is damped to one-tenth of its value in one cycle if 2 p n / w = 2.3. The oscillation can be considered as sharply damped, of course, when the value of Q exceeds 10. E . Pressure Explosion Peninsula and Explosion Limits. Lewis and Feitknechta2 have noted the resemblance between the explosion limits curves for the Ha-02 and CO-02 on one hand and C4H10-02 and Bra-Os on the other. The former group has a lowtemperature explosion peninsula, and the latter has a low-pressure peninsula. The limit curves in the pressure-temperature diagram are of the same shape except that the temperature and the pressure axes are interchanged. When the present kinetic scheme is compared with the kinetic scheme discussed in the unified chain and thermal explosion theory for the H2-02 system, one also finds a close resemblance, the only difference being that the second-order chain termination reaction in the H2-02 system is replaced by a chain termination reaction with activation energy higher than the activation energy of the branching reaction. The system CH4-02 shows both the low-temperature and pressure explosion peninsulas and involves both kinetic mechanisms operative in different temperature and pressure ranges. The pressure explosion peninsula for the hydrocarbon system may be due to a low-temperature branching chain in contrast to the low-temperature explosion peninsula of the hydrogen system which is attributed to branching chain at low pressures. This also can be deduced qualitatively from the present model. Consider the steady-state concentration of the carrier in eq 4
xs =
ki
kti
- k b + ktz
Let us assume that the activation energies satisfy inequality 11 Et1
< E b < Etz
Let us further assume that ktl is independent of pressure (such as the case of wall destruction of carriers) while k b and kt2 are first order with pressure. xs will always be finite a t very low pressures or temperatures as the denominator of x, is dominated by ktl. At high tem-
(18)
Oscillation around a focus with high damping factor
(22) F.B.Lewis and W. Feitknecht, "Bodenstein Band," 1932,p 113; J. Amer. Chem. Soc., 53,2910 (1931). Volume 79,Number 10
October 1969
3400
C. H. YANGAND B. F. GRAY
r\: 0
jx
0 Y*
.I
7s
-
0
x
sure ranges. At the same time, each value must be within the acceptable range determined from other general considerations or measurements, if available. Other thermophysical properties of the reacting mixture (specific heat, heat conduction coefficient, etc.) are estimated. The values employed are
0
Cd
C, = 11N0, cal cc-' OK-'
a -
0
to,
,':
,
500
600
Curve A = AT, curve I3 =
P x
To
IN
OK
X,,curve C
= T,.
peratures, ktz is greater than kb and zswill always be positive and finite. There can be no chain explosion in either case. However, a t temperatures between these two extremes, the value of kb may approach the sum of k t ~and ktz for critical pressures. When this happens, an explosion may be induced. Explosions of this type can indeed be calculated by directly integrating eq 2 and 3. The critical Po and T ovalues for this type of explosion will be shown to form the explosion peninsula in the Po-To plane. The regular explosion limits are obtained by considering the criterion developed in our previous work.la The condition of explosion is A D = BC which in our present case reduces to
[(kbhb
+ ktlhtl +
ktZht2)
x
+ ktz bT b[(kbhb + ktlhtl + htZhtZ)z + d[ki - (ktl
kb'"l]z
bT
zg
T
3
kihi
=
=4cm
PO
N o
mol cc-'
- 760 X 82 X TO'
X
= 1.4 X
'I ktl
= 21.6 for cylindrical vessel =
cal cm-l OK-l sec-l
A tl(No)l~ze-Et'/Rz'
ktz = A tz(No)e-Etz/RT kb = AbNoe-Eb/RZ'
hi
=
Ai(No)2e-E'/RT
(21)
The selected values of the preexponential constant and act'ivation energies as well as the heats of reaction are listed in Table I. Table I : Selected Values of Rate Constants and Heat of Reaction Atl = 3 . 3 x 103/d,
E , ~=
o
mo11/gcc-'/2 sec-1 Atz = 7 . 8 x 1010, sec-l E,%= 1 . 6 X l o 4 Ab = 1.38 X 108, Eb = 7 X lo3 sec-" A i = 1.6 X 1010, E , = 2 . 4 x 104 mol-' cc sec-1
hti = 0
htz = 2 . 0 X lo4 hb = 4.0 X lo3
hi =
o
T,
- L] X 1
The explosion limit curve which relates Po and To is obtained by eliminating xJ and T , in the eq 20 from the steady-state values obtained from eq 4 and 5 .
111. Numerical Calculation of Explosion and Oscillation Limits and Negative Temperature Coefficient To test the theory quantitatively we calculated the explosion limits, oscillation limits, explosion peninsula, and the negative temperature coefficient numerically with assumed rate constants and heat of reactions. The selection of these values is not, however, entirely arbitrary. I t is guided by the criterion that the selected values must produce oscillations and explosions and the experimental observed temperature and presThe Journal of Physical Chemistry
d
X"
800
700
BATH TEMPERATURE,
Figure 1 .
m
Most of the heat of reaction is assumed to have been released in the high-temperature termination reaction. However, the numerical results are not expected to be greatly different if larger portions of heat are released in the branching and in the low-temperature termination reactions. Computation methods and results are discussed in the following sections. A . Negative Temperature CoefJicient. The negative temperature coefficient effect is calculated for a constant pressure = 294 mm. xs and T, values can be calculated from eq 4 and 5 for each given value of To. Let AT = T , - To. Following Norrish and Knox3we take AT to represent the rate of reaction inside the vessel. AT is plotted against bath temperature, TO, in Figure 1 as curve A. At bath temperatures below GOO'K, the reaction rate is very small. This should be identified as the slow reaction zone outside of the cool flame region. At GOOOK, the sudden jump of the rate is due to the surge of the branching rate at this temperature that the denominator of eq 4 becomes very small. The rise
SLOWOXIDATIOX OF HYDROCARBON AND COOLFLAMES of reaction rate followed the rise of the steady-state carrier concentration xs (curve B). The pronounced negative temperature effect occurs between 601 and 680°K. After reaching a minimum at 680°K the rate rises sharply again as Ici starts to rise, which increases both the values of xs and AT a t higher temperatures. The T , and x, calculated here are not all stable nodal points. For T o = 6O1-62O0K, the singularities are stable focuses which are associated with damped oscillations. However, \Then time-dependent solutions a t these focuses are integrated from (2) and (3), results show that the oscillations are very sharply damped. (The minimum damping per cycle calculated is 15.) The maximum amplitude of temperature during oscillation is less than 3°K. Oscillations with such small amplitude generally will not be visually observed. The measured data of negative temperature coefficient by Knox and Norrish3 for the ethane system are also plotted in Figure 1 for comparison (curve D). The steady-state temperature T , plotted in curve C shows a sharp jump at To = 601°K. It maintained fairly constant until the bath temperature passes 640°K. This is consistent with the measurements of Knox and Sorr i ~ h . It ~ can be seen clearly that the high-temperature chain termination reaction lctz is the main cause for the negative temperature coefficient. The heat release rate in the present scheme is directly proportional to the carrier concentration xs. When the high-temperature termination reaction becomes effective at bath temperatures above 601°K both x, and heat release dlwease to exhibit the negative temperature effects. The calculated negative temperature coefficient effect is very sensitive to the initial pressure. Strong negative temperature coefficient effects occur at pressures immediately below the oscillation limits. Figure 2 shows negative temperature effects calculated for different pressures. At a pressure of 294 mm, a sharp negative temperature effect is shoxm. At Po = 280 mm the effect is milder but still significant. The heat release rate rises up to a maximum a t TO = 615°K. Knox and Norrish’s measurements (plotted in Figure 1) resemble this curve more closely than does the one for a pressure of 294 mm. At Po = 260 mm, the calculated curve in Figure 2 shows no negative temperature effect. The flat portion of the curve betlveen To = 635 and 680°K is sometimes referred to as the zero temperature coefficient. At even lower pressures the slope of the curve becomes always positive. Bardwell and Hinshelwoodj observed, however, an inflection point is always present5on this curve within the same bath temperature range (640-680°K). B. Theymokinetic Oscillation. It is pointed out in section I1 that the regime where oscillation is possible should be enclosed by a curve in the Po-To plane prescribed by eq 15. It seems straightforward t o solve for this curve by solving eq 15 in conjunction with eq 4 and 5 which prescribe the values of x, and T,. This
3401
ot
500
6 00
700
BATH TEMPERATURE
8 00 To I N O K
Figure 2.
\
800-
E 750E
z -700Y
a c $650Y
I
Y
c
600
-
550-
>
200
300
400
FIGURE 3
600
500
PRESSURE
IN mm
Figure 3.
procedure is cumbersome, however. A program has been developed to solve eq 4 and 5 for T , and x, for given pair of values of Po and To. The nature of the singularity corresponding to T , and xs is then tested. Inequalities 13 and 14 are used to locate the curve prescribed by eq 15. At the singularity the damping factor p, period of oscillation, and damping per cycle (Q defined in section 11) are also evaluated. The whole Pa-To plane is mapped until explosion and oscillation limits can be determined within the accuracy of 3°K for a given pressure. Pressures are calculated a t increments of 10 mm or less. The limits prescribed by eq 15 are plotted in Figure 3 as the closed curve EFGH. Adjacent to this curve are all stable nodal points. Inside EFGH, however, the regime is divided into two parts by the curve LXK. The area enclosed by the curve LNKC has unstable focuses where p is positive. The area enclosed by curve EFKNLGH has stable focuses where p is negative. Oscillation may not be observed everywhere inside the curve EFGH. Area covered by the curve BCD is the explosion peninsula where explosion instead of oscillation will be observed. This will be elaborated in the next section. The traVolume 73, Number 10
October 1969
C. H. YANGAND B. F. GRAY
3402 Table I1
W,
To,OK
560
570
680
690
600
610
620
630
640
650
660
oscillation period, sec
2.2
2.4
2.6
2.6
2.8
3.0
3.2
3.4
3.8
4.2
6.0
Table I11 To,"K
560
570
580
590
600
610
620
630
640
650
660
0.74
0.78
0.98
1.1
1.3
1.75
2.15
3.2
5.85
15
174
n, damping per cycle
jectory solutions of eq 2 and 3 will show that the oscillastarted with a peak temperature of 810°K after an tions inside the area enclosed by LNKC are about stable initial induction time of 29 sec. (The induction time limit cycles where amplitudes are large. Oscillations here is only a part of the total induction time. It will be discussed later that another part of the induction in the area to the left of the curve MNK are about time will be the period for the accumulation of aldestable focuses. Amplitudes, however, are very rapidly damped as temperature increases. Most of the oscillahydes in alkane-oxygen mixtures.) It is then damped tions are centered in the area below the explosion pento a stable limit cycle very rapidly. The integrated curve B at To = 580°K is obviously insula and close to the curve MNK as observed in the classical work of n'ewitt and Thornes,' Day and Peasej2 about a stable focus. The induction time is much less than curve A being only 4-5 sec. The period of oscillaKnox and N o r r i ~ h ,and ~ , ~Bardwell and HinshelwoodP tion is 3.1 sec, compared with the estimated value of The calculated period of oscillation and damping factor 2.6 sec in Table 11. I t is interesting to note that the per cycle at P = 480 mm are shown in Tables I1 and peak temperature a t To = 560°K is about 60°K higher 111,respectively, at various bath temperatures, To. than at To = 580°K. It is further reduced for the case I n Tables I and I1 the periods and damping factors (curve C) To = 640°K. At To = 640"K, the system of oscillation are estimated with a linear approximation still oscillates about a focus as we can see from Figure near the singularity. These values when compared 3. The point PO = 480 mm and To = 640°K is well with numerical integration give surprisingly good agreewithin the closed oscillation limits EFGH. The inducment. The experimentally observed values of oscilla~ ~ ~time ~ ~ for this case is even shorter, about 2-3 sec. tion period are of the order of a few s e ~ o n d s . ' ~ tion The amplitude is damped sharply. Oscillations can For P = 480 mm, three trajectories are integrated and hardly be detected. However, the computer output plotted in Figure 4 a t To = 560 (curve A), To = 600 shows clear oscillatory behavior. (curve B), and To = 640 (curve C). At To = 560 the The computed wave form of oscillation is always of system oscillates about a stable limit cycle. The averthe type of a pointed spike that is followed with a wide age period of oscillation is 2.7 sec. The oscillation valley. This seems to be consistent with experimental observations of Bardwell and Hin~helwood.~ The oscillations a t the points Po = 240 mm, TO = 590°K and Po = 340 mm, To = 606°K are plotted in Figure 5. An examination from Figure 3 shows that these two points are unstable and stable focuses, respectively. The oscillation about the unstable focus again converges to a limit cycle with smaller amplitude than the previous case shown in Figure 4. The induction period, however, remained about the same while the period increased to 5 sec. The fuel consumption in the present model is neglected. The calculated oscillations, therefore, become perpetual oscillations of the limit cycle. I n an actual case, the oscillations will be terminated when the fuel I I 1 I 500 is used up. More cycles of oscillations are expected to be observed when the periods are comparatively short, thus entailing smaller fuel consumptions per cycle. The Journal of Physical Chemistry
SLOWOXIDATION OF HYDROCARBON AND COOLFLAMES
3403
P0=340
SO01
I 10
I 20
I
30
I 40
I
SO TIME
I
I
60 70 I N SECONDS
I
80
I
500
10
Figure 5.
I
TIME
1
I
20
30
IN
40
SECONDS
Figure 6.
C. The Calculation of Explosion Limits and Explosion Peninsula. The normal explosion limits are calculated in the same fashion as the oscillation limits. For a given pair of values of Po and To,eq 4 and 5 are solved for singularities. The Po and To values in the regime between the curve ABD and EHG always give two singularities, a nodal point and a saddle point. These two singularities merge into a saddle-nodea3on the curve ABD. ABD satisfies eq 20 as the explosion limits of the ~ystern.'~The region above the curve ABD contains no solution for singularities of eq 4 and 5. I n this region then, the steady states do not exist and the region is unstable. The explosion curve BCD does not follow the explosion criterion established in the unified theory. (Trajectories of self-oscillation and explosion are studied in the phase plane. Results are to be published separately.) Equations 1 and 2 must be integrated with the bath temperature as the initial value for temperature and zero for the carrier concentration. Such integrations below the curve BCD have yielded oscillatory trajectories plotted in Figure 4 and discussed in the last section. Integrations at the points PO = 550 mm, TO= 550°K and PO = 550 mm, To = 580°K are shown in Figure 6. These two points are on each side of the BCD curve that envelops the explosion peninsula. Instability occurred after induction periods of 35 sec for TO = 550°K and only 5-6 sec for TO= 580°K. Integrations inside the entire explosion peninsula shown the same characteristics. The difference in induction time on each side of the peninsula has been observed by many investigators. The curve IFD in Figure 3 can be regarded as the explosion limits calculated with the pure chain theory, The curve is obtained by letting 9124,25
kb
- ktl - kt2 = 0
the important oxidation characteristics are enveloped by this curve. It is an indication that all these characteristics are associated with the autocatalytic chain process.
IV. The Autocatalytic Carrier and the Kinetics of Hydrocarbon Oxidation Frank-Kamenetski? and Salnikoff's'l theoretical work on cool flame oscillation has stimulated interest in the identification of the active intermediates which cause the cool flames. The literature is divided on peroxidess and aldehydes3 or both.' Both of these species are found to be intermediates during the oxidations of most hydrocarbons. However, conclusive arguments or evidences are not established for either one or both together, being the key intermediates. Knox and Norrish's argument ruling out peroxides as the important intermediate is based on the fact that cool flame oscillation is observed for the ethane system3 where peroxides are not found in measurable quantities. Subsequently, cool flames have also been observed4 for methane where the situation is quite similar. They have reaffirmed the belief that the higher aldehyde is the critical intermediate which was suggested earlier by Newitt and Thornes.' In his study of propane oxidation, ShternZ5 made the following interesting observation. At 300 and 350", addition of CH3CH0 to a C3H8 0 2 mixture shortens or suppresses completely the induction period (suppression occurs when CHsCHO is added in amounts corresponding to its maximum concentration). But if CHsCHO is added a t 387" (where a zero temperature coefficient is observed), it has no effect on the reaction. He concluded that CH3CHO is the active intermediate for the
+
(22)
which predicts explosion limits in the pure chain theory. It is clear that the I F D curve does not prescribe limits in the present system, but it is interesting to note that
(23) A. A. Andronov and 5. E. Khaikin, "Theory of Oscillations," Pergamon Press, New York, N.Y., 1966. (24) M.Vanpee, Compt. Rend., 243, 804 (1956). (25) V. Ya. Shtern, Zh. Fiz. Khim., 28, 613 (1954). Volume 78, Number 10
October 1969
C. H. YANGAND B. F. GRAY
3404 propane system. The present analysis is essentially in agreement with the argument that higher aldehydes are the important intermediates for oxidation of hydrocarbon with two or more carbon atoms. For the methane case, it is formaldehyde. Even though aldehydes are critical intermediates in hydrocarbon oxidation, they should not be identified as the autocatalytic carrier in the present model. Cool flame oscillation and pressure explosion peninsula are observed in oxidation of acetaldehyde, propionaldehyde, and isobutyraldehyde.26 There is no obvious reason to assume that the autocatalytic carriers are different for aldehydes and saturated hydrocarbons. The general scheme postulated here for hydrocarbon oxidation is that aldehydes are accumulated during the induction period. When the aldehydes reach significant concentration, a branching reaction starts to accelerate the oxidation process. The aldehyde accumulation reaction, the sinking and branching reactions of the autocatalytic carrier, and the mechanism of cool flame radiation will now be discussed. A. Aldehyde Accumulation in Hydrocarbon Oxidation. The accumulation of formaldehyde in the oxidation of methane is generally accepted to be
+ CHI + HOz + 02 HCHO + OH OH + CHI +HzO + CHI
CH, CH,
0 2 ----jc
initiation
(23)
propagation
(24)
--t
The accumulation of acetaldehyde, propionaldehyde, and butyraldehyde in the ethane, propane, and butane systems may go through a similar scheme. However, the mechanism of accumulation of higher aldehydes from higher hydrocarbons is somewhat different from that of CHzO from methane and, in general, faster. The application of gas chromatography to product analysis has shown that the primary oxidation product above 300" is, wherever possible, the conjugate olefin.14 This, of course, is not possible for methane. Knox14 has proposed that the double bond in the olefin is then split to give two carbonyl compounds. For example, ethane would give two molecules of formaldehyde by this scheme, whereas it would produce one molecule of acetaldehyde by the scheme (presumably much slower) similar to (24). Hence, in ethane, the branching will be mainly due to CHzO and will only occur at higher temperatures and in larger vessels. Knox's scheme for accumulation of CHgO from ethane would be CzHs R +CzHs RH CzHs 0 2 ---t CHz=CH* HO2 CHz=CHz 2H02 +2CHz0 20H OH CzH, +CzHs HzO (25) The simplest hydrocarbon which can produce acetaldehyde by Knox's scheme is propane. The production of acetaldehydes can be expected, and the production of propionaldehyde also becomes possible in the
+
+
+
+
The Journal of Physical Chemistry
+
+
+
+
OH f HCHO CH02
3 HzO + CHO ,branching
+ HCHO -% HCOOH +
/,
COz
CHO
+ Hz
(26d) We)
(26) C. M. Newitt, L. M. Baxt, and V. V. Kelkar, J. Chem. SOC., 1703 (1939). (27) M.Vanpee and F. Grard, Fuel, 34,433 (1956).
SLOWOXIDATION OF HYDROCARBON AND COOLFLAMES CHO "tt wall destruction CHO
+ R4 -% CO + H + M
(26f) termination
'
(26g)
lar to the above case
O2
+ CH3CH0 -% HOz
+ CH3C0
+ O2kb'_ CH3CO2 + 0 0 + CH3CHO "bC OH + CH3C0
initiation
1
+ CH3CHO 2H20 CH3C0 + 1 branching
1
CH3C02
+ CH3CHO -%
CH3COOH CH,
c+ COz
+ CH3CO
ktl
CH3CO +wall destruction CH3C0
+ RI -%- CHa +
co + AI
CHI
1 .termination
+ 02-% CO + HCHO+OH
1
Reactions 26g and 27g are probably followed by reactions that produce OH radicals
+ + AI e HO, HOz + HOz * 20H + CH3 + HCHO + OH H
0 2
0 2
(28) The OH radicals will naturally attack the saturated 0 2 --t
hydrocarbon to replenish the aldehydes which were consumed in the branching chain. Reactions 26g and 27g by themselves are endothermic. However, they may be regarded as exothermic if the thermicities of the reactions that follow them are added. The activation energies of the branching and high-temperature termination reactions for CHaCO, CzHsCO, and C3H7C0 radicals are probably very close in view of the work of Newitt, et al. In the case of oxidation of pentane, acetaldehyde, propionaldehyde, and butyraldehyde may all be accumulated in sizable portions a t early stages. Since their chain mechanisms are operative in the same temperature range, they can be represented in the scheme of single chain with the sum of the concentrations of all three species. C. Cool Flame Radiation. The emission spectra of all cool flames are identical with the fluorescence spectrum of formaldehyde,28 but the quanta emitted per unit volume is small in comparison with fuel molecules reacting with a ratio2e*30 of to 10-l6. The reaction in which the excited formaldehyde is formed must be exothermic to the extent of at least 77 kcal/ such circumstances, mol, according to W a l ~ h . ~Under l the following radical reactions, even though infrequent occurrences, should be considered H
+ CHO -+
CHCO*
CH3CO
OH
3405
HCHO*
HCHO
+ hv
(29)
The C-H bond dissociation energy is known to be 87 kcal. The near degeneracy of the vibrationally excited ground state with the electronically excited state makes the radiative recombination reaction probable with emission through its fluorescence band when the temperature of the reacting medium is in the neighborhood of 700°K where the CHO radical concentration is appreciable. Cool flame mechanisms for the CHaCO and CzH7CO radicals are unlikely without the similar degeneracy of the excited states. The temperature oscillations in either the ethane or the propane system induce the cool flame radiation whenever the peak temperature reaches the level where the CHO concontration is appreciable. The temperature inside a reaction vessel is not uniform. During a cycle of oscillation, the peak temperature must be reached at the center of the vessel first on account of the heat conduction process. The cool flame emission, therefore, will be more intense in the center of the vessel due to higher temperature. As the temperature of the cooler gas surrounding the vessel center gradually rises, the intense radiation propagates (28) W. J. Levedahl and H. P. Broida, Anal. Chem., 24,1776 (1952). (29) J. E. C. Topps and D. T. A. Townend, Trans. Faraday SOC.,42, 345 (1946). (30) W. G. Agnew and J. T. Agnew, Ind. Eng. Chem., 48, 2224 (1956). (31) A. D. Walsh, Trans. Faraday SOC.,43,305 (1947).
Volume 78, Number 10 October 1968
C. H. YANGAND 13. F. GRAY
3406 to the edge of vessel as if a cool flame is spreading from the center. The contrast between the cool flame and the uniform luminosity in the vessel, therefore, represents the local temperature gradient which is a function of the amplitude of the temperature oscillation. From curve C in Figure 1 we note that the steady-state gas temperature T , increases with bath temperature and from the trajectory calculations plotted in Figure 3, the amplitude of oscillation is shown decreasing with bath temperature. It is expected, therefore, that the luminosity and the cool flame, naturally, will decrease with the bath temperature. The oscillation amplitude along curve EH approaches zero. Thus, no cool flames can be observed along this curve. At this temperature range of the Po-To plane, the cool flame zone and the uniform luminosity zone will not be divided by a distinctive boundary as the amplitude of oscillation approaches zero asymptotically on curve EH. Along curve CFK in Figure 1, sharp characteristic changes are encountered owing to bifurcations. It is expected to be a distinctive boundary of oscillation with high contrast between the luminosity and cool flame as the amplitude of oscillation is large, and T , is comparatively low. The experimental observations of Newitt and Thornes' fit this behavior. Chernesky and BardwelP observed weak oscillation without cool flames in their experiments of butane oxidation. They coated the reaction vessel with PbO. The absence of the cool flames is probably due to the contrast of the cool flames which is too small to be observed in a weak oscillation with small temperature amplitude.
V. Discussion The present mathematical model and proposed kinetic scheme have provided a close characterization of some of the important experimental observations of hydrocarbon oxidation. The selection of the numerical values of the rate constants, may, at a first glance, appear to be completely arbitrary, but when it is demanded that the computed explosion limits, oscillation limits, oscillation periods, explosion peninsula, etc.. all be consistent with exDerimenta1 values, the apparent flexibility of the rate constants becomes constrained. A variation of the activation energy for ktz of more than 20% of the presently assumed value will make it impossible to have the calculated limits fit the experimental values, no matter how the rest of
The Journal
of
Physical Chemistry
the parameters are numerically varied. It is significant to note that the determination of a set of rate constants for a complex chemical system can be quite precise if a large number of complex characteristics are to be accounted for simultaneously; such is the present case. Other specific points relating to the model and the proposed kinetics are discussed below. A . Even though the proposed reactions for the kinetic scheme seem to be quite plausible, there may be other alternatives that also fit the scheme. Thus the identification of the herein discussed reaction schemes can only be regarded as tentative. B. The branching reaction using aldehydes as intermediates was proposed by Knox and Norrish4 in their work on the oxidation of propane RCHO
+
0 2
+ [RCOOOH] +RCOOH
+0
This reaction, however, is not autocatalytic. The chain reaction which we proposed, on the other hand, is autocatalytic with RCO as the carrier. It contains the following two steps
+0 RCOz + RCHO +RCOOH + RCO RCO
+
0 2
+RCOz
It is interesting to note that they reduce to the same over-all scheme proposed by Knox and Norrish. C. The termination rate constant lctl is assumed to be one-half order in the pressure in the present calculation. If a zero-order rate is assumed the computed limits will more closely resemble the results of Pease and Day.2 Both the Newitt and Thornes' and Knox and Norrish4 results are, however, more closely simulated by the present results. D. The methane and oxygen system can be investigated with a similar scheme using higher activation energies for the branching reaction and the termination reactmionkt2. The double explosion peninsula and delayed ignition phenomena appear to be explained readily by superimposing the branching mechanisms for formaldehyde and the higher aldehydes (acetaldehyde, propionaldehyde, and butyraldehyde) which is presently under investigation. Acknowledgment. The authors are indebted and grateful to Professor A. L. Berlad of the State University of New York and to Dr. J. Griffiths of the University of Leeds for many discussions and invaluable suggestions.
