2212
0. R. RICE, AUGUSTINE0. ALLEN AND HALLOCK C. CAMPBELL [CONTRIBUTION FROM THE CEiEMICAL
VOl. 57
LABORATORY OF HARVARD UNIVERSITY]
The Induction Period in Gaseous Thermal Explosions1 BY 0. E;. RICE,'^ AUGUSTINE0. ALLEN AND HALLOCK C. CAMPBELL We have recently studied the explosions of azomethane2a and ethyl azidej2' and have shown that they are probably thermal explosions, due to self-heating of the gas on account of the exothermicity of their slow decompositions. In these papers we have compared our results with the theory of S e m e n ~ f fwhich ,~ gives the critical explosion pressure as a function of the temperature. Now it is observed when the gases are admitted to the reaction vessel that an appreciable time elapses before the explosion occurs. These lag times or induction periods have been measured in many cases, and in this paper we shall present these data, and discuss the theory. We shall see that this enables us to make an estimate of the heat of decomposition. The basis of the theory which we shall present consists in following the change of temperature of the gas as reaction proceeds. An equation giving the amount, T , by which the temperature of the reacting gas exceeds that of its container as a function of the time and from which the induction period can be obtained has been given by Allen and Rice.2a The rate of production of heat is given by the expression QknV, where Q is the heat of reaction per mole of gas decomposed, k' the volume of the reaction vessel, n the number of moles of reacting gas per unit volume, and k the rate constant. k = "), A being a constant, R the gas constant? E the activation energy for the unimolecular decomposition, and T 7; the temperature of the gas; TOis the temperature of the reaction vessel. We may where no is the initial value of n write n = noeWkT, and t is the time, provided we assume as an approximation that k is constant during the induction period. Heat is removed from the gas by conduction and convection a t the rate axT, ( L being the wall area and x a constant. Subtracting the rate of loss of heat from the rate of production and dividing by C, the total heat capacity of the gas in the vessel, we get the rate of +
-+
(1) Presented a t t h e New York Meeting of the American Cheinicnl Society, April. 1935. (la) Present address, University of Califurnia, Berkeley, Calif. (2) (a) Allen and 0. K. Rice, THISJOURNAL. 87, 310 (1936); (b) Campbell and 0. K. Rice, ibid., 67, 1044 (1935) (3) Semenoff, Z . P h y s i k , 48, 3 7 1 (1928)
change of the temperature of the gas; making a slight reduction we obtain the equation dT/& = e - E / R ( T f To)e--k(C/Bb - (ax/B)T (1)
where B = QAnoVand r = tB/C; here t is measured from the moment the gas is admitted to the vessel, assuming that it warms up instantly to the temperature To,an assumption that will be discussed later. Since C as well as B is proportional to no, we may, for any given series of runs involving a definite substance and reaction vessel a t a definite external temperature, To, but with varying pressure, consider T to be a measure of the time from the start of the experiment expressed in a special set of units. Likewise the quantity B/ax, assuming that x is independent of the pressure, is a measure of the pressure used in a given experiment] and its value determines whether or not an explosion occurs in an experiment performed a t a given temperature, To. In a case in which an explosion occurs integration of (1) will show that the temperature rises very suddenly after a lapse of time, and the time at which this sudden rise occurs may be set equal to the induction period,4 while if explosion does not occur the temperature merely rises to a maximum and falls off again. The differential equation (I) cannot be integrated analytically, but it may be solved numerically by the use of the Runge-Kutta formula.5 In order t o compare with the experimental data it is necessary to go through this process for different values of T , and nx/R. When this is done the value of T at which the explosion occurs is known for any given conditions; this may be compared with the observed induction period t and B!C found. Now B/C = QAnnV/mVChf = QA/ChI where Cbf is the heat capacity per mole of azomethane; as A is known, we can calculate @IChf which should be practically constant for all experimenls with a given composition of gas. If there art. r nioles of inert gas per mole of azo( 4 ) T h e same pririciples h a r e been used by Tizard and P y e [Phil. M\.inn., 44, 79 (l922)l in ciisciissing t h e oxidation of certain hydrocarbons, hut their enlw-irnents were of a different type t h a n ours, and their equations cannot he used in treating t h e experiments of t h e type considered here. (.i) Scarhorourh, "Xurnericnl Mathematical Analysis," T h e Johns Hopkins 1.nivcrcity Press, p. 273.
Nov., 1933
THEINDUCTION
PERIOD IN
GASEOUSTHERM-4L
2213
EXPLOSIONS
+
methane, then Cl,I = CA rCc, where CA is the molal heat capacity of azometliane and CG that of the inert gas. In order to take into account the correction factore - k ( C ! B ) ~, which allows for the amount of gas decomposed before explosion takes place, and which affects the induction period near the explosion limit and also changes the value of the critical explosion pressure somewhat, it is necessary to assume a preliminary value of C / B , which is precisely the quantity we are trying to obtain. One is thus required to make a series of successive approximations, a very long process on account of the tedious calculation involved in the use of the 1 I I 1 I 1 I Runge-Kutta €ormula. This fact, together with the desirability of having an easy way to perform I the integration for a considerable range of values of the constants involved, has led us to seek an approximate method of calculation. This will be presented in $2. It is first necessary, however, to carry out the more exact integration of Equation (1) for some special values of the constants involved, in order to gain some informa--tion about the general character of the solutions 20 of (1) and to be able to verify the accuracy of the approximate method. --------.An 3.87 For this purpose we have taken among others ithe case where To = 630°K. and k C / B = 1.27 x and E = 51,200 calories per mole. These values were estimated to correspond roughly to the situation with azomethane a t ti3O0K. Figure 1 shows the calculated T US. T curves for various values of ax/B. It is seen that, as expected, the curves are definitely of two 2 4 6 8 10 classes, the curves for small values of ax/B (large x 10-19. values of the pressure) being such as one would Fig. 1.-Theoretical temperature time curves at 630°K. for various values of (Ar):B( X 10~9). expect to correspond to an explosion, while the curves for large values of u x / B correspond to a In spite of the probable failure of the theory for pressures quiet decomposition in which the temperature very close to the explosion limit it will not be without inreaches a relatively low maximum, and then terest to examine a little more closely the nature of thr drops off. In the case of the explosive curves, solutions of the differential equation (1). Every one of the theory does not predict an exact instant for the T vs. 7 curves will ultimately reach a maximum, even though it be of lhe explosive type. This maxinium will the explosion to occur, but certainly fixes it within necessarily lie on the locus an interval of time much smaller than the experi- ( k C / B ) 7 = - E / R ( T + TO)- ln(ax/B) - In T (2) mental error in determining the induction period, which makes d T / d r , as given by Equation (l),zero. The The critical value of ax/B appears to lie between locus defined by Equation ( 2 ) for a x / B = 2.86 X 10-19is 2.86 X and 2.57 X lo-*$. It is, however, shown (broken curve) in Fig. 1; the loci for other values not certain that this value can be determined as of a x / B near this will be of just the same shape, but disclosely as indicated by these figures, for very placed along the 7 axis a short distance. These loci are for T = 0 and eventually s-shaped, starting at T = close to the explosion limit the curves are very reaching 7 = - m for T = m . There are two points sensitive to slight inaccuracies in the calculation, where the locus (2) has a vertical slope; one is shown, and it is quite possible that one which appears to the other one is at a value of T approximately equal to
!I i
r
/ -
-I1
n,,
0. K. RICE, AUGUSTINE0.ALLENAND HALLOCK c. CAMPBELL
2214
is of the order of 25,000", and a vdue of r approximately ( B / k C ) @/ET0 1 2 la(RTQ/E)),or about 32 X 10lg,greater than the value of r for the f i s t point of vertical slope. This value of r gives the longest theoretically possible induction period. The explosive curves theoretically reach the upper branch of (2), that is, the part of the locus defined by (2) which is going out to r = w ; the valyes of T thus theoretically attained are greater than 25,000'. Such values will obviously not actually be reached, for one reason because k will not actually be independent of T and therefore the azomethane is used up before such temperatures are reached. The curves which have their maxima on the lower and middle branches of (2), up to the second point of vertical slope, where the locus defined by (2) turns around and heads toward r = -a, are the non-explosive curves. Theoretically there is a perfectly continuous transition between explosive type and non-explosive type curves, the transition occurring in an astonishingly small range of values of ax/B. E / R , which
-
+
-
$2, Approximate Integration of Equation (1) In order to get an approximate integration of Equation (1) we first see what can be done if the term e-k(C/B'r is set equal to 1. Under these circumstances we may, from Equation (I), write for T ~ the , critical value of 7 a t which explosion takes place
The upper limit in this integral is obtained in the following way. In the theory of Semenoff the rise in temperature at the explosion limit is equal approximately to2a RTo2/E, this being also very approximately the temperature a t which the locus defined by Equation (2) has infinite slope, as shown in Fig, 1. From Fig. 1 it is obvious that, though T, cannot, as we have said, be exactly defined, it will be very reasonable to determine a value by integrating to the temperature where T is equal to twice the value RTo2/E. If we make the allowable approximatione T/TO)-' = 1 - T/To then we may write (1 Equation (3) in the form
+
7c
=
SOBRToVE
[e--E/RTo eET/RTo'
- ( u x / B ) T ] - ld T (4)
(6) We may note t h a t under this approximation Equation (1) may be written in t h e form
dy/dJ = euwhere
1
e-+J
- fy
-
J = r(E/RToe)el E / R T Q = k ( C / B ) ( R T o 2 / E ) e E / R T Q- 1
+
and Y and I are defined just following Equation (4). This would make t h e Rung-Kutta calculations much easier, for t h e effects of To and k ( C / B ) are both taken care of essentially by t h e one parameThis would greatly lessen t h e number of calculations needed, ter but they are still su5ciently laborious t o make i t seem worth while t o develop t h e approximation method of this section.
+.
VOl. 57
If we set y = BT/RTo2andf = (ax/B) (RT02lE)
e E / R T ~- 1
this takes the form
ro = (RTo2/E)e-E/RTo- 1 lo
where Io =
( e ~
1
(5)
- fy)-1 dy
(6)
IOmay be evaluated as a function off from Equation (6) by numerical integration. By its definition, f is inversely proportional to P,the pressure, x being assumed independent of pressure. Now the equation for the critical explosion pressure' is also based on the same approximations made so far in the considerations of this section. Using this equation we see that at the critical explosion pressure f is equal to 1 and we may set f = P*/P
(7)
where P* is the critical explosion pressure of the approximate theory. For explosive runs, of course, j < 1. Allowance for the term e-k(C/B)Tof Equation (1) may be made in an approximate manner as follows. In the actual calculation of the curves in Fig. 1 one finds that the value of 7 a t which the curve begins to rise rapidly is largely determined by its behavior in the neighborhood of the inflection point, which occurs a t a value of T very near to 7,/2 and a value of T very near RTo2/E. This suggests that we replace the exponential term in question by a sort of average value, namely, e-k*(C/B)T0/2, where K* is the value of the rate constant at the temperature TO RTo2/E. We then get in place of Equation (4),the equation
+
r0 =
2RTo'/E
[e-E/RTo eET/RTo'e-k*(C/E)s,/l
-
(ax/B)TI-' d T
(8)
in which T~ is involved inside the integral sign as well as on the left-hand side of the equation. This equation is to be solved for 7,. It is convenient to define a quantity I , similar to the quantity IOof Equation ( 5 ) by the relation ro = ( R T o z / E ) e E / R T o - 1 I (9) the difference between the value of T, obtained from Equation (8) and that given by Equation (4)being expressed by the difference between I and 10. If we make the substitutions following Equation (4) and, further, set 8 = k*(C/SB)(RT,Z/E)eE/RTo - 1
(10)
then it is readily seen from Equations (8) and (9) that I = e91
1
(er
(7) Ref. 2a, Equation ( 5 ) .
- 1 - eerjy) -
dY
(11)
THEINDUCTION PERIOD IN GASEOUS THERMALEXPLOSIONS
Nov., 1935
Now the integral appearing in Equation (11) is simply the integral IO with the argument fee' appearing in place of f. We can thus write a functional equation connecting I and Io
IV,
=
eeICn IoocfeeW)
(12)
2215
ing to the particular value of 7,. It is seen that the agreement is very good, especially when one bears in mind that the Runge-Kutta method does not give the explosion limit very accurately, and that all points of a given set of calculations should be shifted horizontally by an amount corresponding to the error in the explosion limit. We may therefore use the the approximation developed in this section with some confidence, except very close to the explosion limit where all calculations break down. Since all our calculations do break down near the explosion limit it is not surprising
in which the quantity in parentheses following an I or Io is the argument of which that I or Iois a function. This functional equation for I may be solved and I found as a function o f f by the method presented in the Appendix. The results of these calculations for a number of different values of 0 are shown in Fig. 2. (The values given on the curves are 0 1 2 3 4 5 6 7 8 9 1 0 20 1760 0.) I 30 Now the fact that the curves bend back indicates l.6 40 that for any given value of 1.48 there is a certain largest value off above which there are no real solutions of the 3 1.2 functional Equation (12). This suggests that this larg1.0 est value of f corresponds to the corrected explosion o,8 limit. Equation (7) gives f in terms of P and P*,the I I I latter being the uncorrected 0 05 0.10 0.15 0.20 h g f . critical pressure. If we wish Fig. 2. to get I for any 8 as a function of PIPI*where PI*is the corrected critical that an attempt to go beyond the explosion limit pressure we can do this by noting that8 results in the upper branches of the curves of Fig. 2, which can have no physical meaning. PIPl* = felf (13) In an experiment we have a measurement of where fe is the largest value of f for the given 8. the time lag, which we may call tc and which we Thus, to get I as a function of PIPI*or fe/f we may assume gives us (C/B)rc. Furthermore, we simply move the curves of Fig. 2 to the left till the know the ratio of the pressure to the critical prespoint 'of infinite slope touches the axis, log f = 0. The result is shown in Fig. 3. The real justifica- sure, which gives Pl*/P. We also know To and tion of this procedure is that in special cases the k*. We therefore choose from among the curves curve obtained agrees with the result obtained of Fig. 3 that one which gives us the correct value by integrating Equation (1) directly, using the of (C/B)rc = 28I/k* (see Equations ( 9 ) and (10)) Runge-Kutta formula. This is shown in Fig. 3. at the given value of PI*/P.We thus find the The circles and triangles represent the values value of 8 corresponding to the particular experiment, and from this we can get Q/CM, where C, T,/ (RTo2/E)eE/R To - , where rc has been calculated by means of the Runge-Kutta formula from ( l ) , is the heat capacity of the reacting gas per mole and PIPI* has been set equal to the critical value of azomethane. This quantity is obtained from 8 by noting that k* = Ae-E'R(To+ R T o P / E )which , is of A x / B divided by the value of A x / B correspondAe' E/RTo. Substituting approximately equal to (8) Equation (13) can be used to correct the critical pressure given this into Equation ( l o ) , and remembering the by the Semenoff theory. However, Equation (14) shows that 8 should be practically constant for any given explosive reaction over definitions of C and B we get
the range of temperatures in which it can be studied. The correction to log P* is therefore essentially a constant independent of the temperature, and hence is of no practical importance. However, for purposes of notation, it is assumed hereafter in this paper that the experimental critical pressure gives a measure of PI*.
€3 =
(c,/Q)(RTo2/2E)
(14)
Since Q/CM will not be expected to vary very much over the range of temperatures which can
0. K. RICE,AUGUSTINE 0. A L L E N
2216
Fig. 3.-Data
.4ND
Vol. 57
HALLOCK c . CAMPBELL
E = 51,200, for Runge-Kutta points: 9, E = 51,200, TO= 630”K., 1760 0 = 3.66; 0, = 6.66. A,E = 41,500, TO= 553’K.. 17600 = 7.82.
TO= 614”K., 17606
be studied, we should get the same value for all experiments with a given substance or mixture, and this will be a test of the theory
53. The Heating of the Gas In the above discussion we have neglected the time it takes to heat up the gas when it enters the reaction vessel. It is rather difficult to make an exact calculation of this quantity because the gas enters the reaction vessel through a hot tube where it receives a preheating, in what is probably a negligible length of time. If this heats the gas to a temperature equal to TO - T’, then the time (multiplied by B/C)for the gas to heat up to the wall temperature, TO,is, from Equation (1) (which should hold for this process as it does for the heating up of the gas after the temperature TO has been reached) 7’
[ e - E / R ( T f To) -
= J:T,
( a x / B ) T ] - ’ dT (15)
where (er - 1
- jy)-I
dy
(17)
where T* = RTo2/E. We have evaluated I’ for a number of values off for T‘/T* = 5 and for T’/T* = 10. These values are given in Table I together with values of IO.It will be seen from this table that I‘ is not very sensitive to T‘/T* when the latter is as great as 5, and all our subsequent calculations have been carried out using this value. TABLE I VALUESOF Io AND I’
f 0.99 .98 .96 .95 .94 .90 .85 .80 .70
IO
I’(T’,’T* = 5) I’(T’,,T”
= 10)
40.81 27.22
3.091
18.80
...
...
3.184
3,914
3.317 3.464
4,087
....
14.82 10.89
...
...
3.791 ...
...
.... 4.280 7.04 ... ... In this we have set the term e-k(C’B)7equal to 1, 5.392 4.010 ... which is surely reasonable in this case. T’ deter.50 3.819 5.152 ... mines the time which elapses before “zero time,” that is before the time a t which the gas is at the $4. Comparison with Experiment temperature To,and it is to be added to rc before Tables I1 and I11 show all measurements of comparing with the experiments. Making the the induction period with azomethane and all of Same substitutions as were made to get T, in the those made with ethyl azides No. 5 and No. 8,!’ form ( 5 , we find (9) See the discussion by Campbell and Rice, Ref. ab, of the variTi
= ( R T i , Z / E ) ~ E / R T-a
1
I’
(16)
ous samples of azide used.
Nov., 1035
THEINDUCTION
PERIOD I N
with the exception of some in which the temperature was uncertain. These runs are arranged in TABLE I1 INDUCTION PERIODS FOR AZOMETWNE EXPLOSIONS T","p.,
C.
Expt.
Press. (total),
mm.
Ind. period, sec.
100% (CHa)*N?200-CC.bulb
255 253 248 247 244 245 242 158 157 156 153 155 154 151 150 147 148 144 146 145 141 142 140 265 263 261 260
341.0 341.0 370.0 370.4 358.3 358.3 357.7 371.9 371.9 371.5 346.8 347.0 347.2 353.3 353.3 386.5 385.7 378.2 378.5 378.5 363.5 363.4 363.6
189 193 30 31.5 54.5 56.5 55.5 27 29 32 101.5 103 104.5 66 68 17.5 19 22 23 24.5 37.5 38.5 41
100% (CH3)tNt 50-CC.bulb 353.0 149 353.5 152 355.3 132 355.0 137.5
m
7.5 m
2.0 m
3.8 4.0 m
2.2 1.4 m
5.2 4.8 m
5.5
173
59.2% He, 221 223 222 218 217 213 215 214 212 210 209 207 206 205 203 202 201 200 199 190 192 191 187
40.8% (CHJ)& 200cc. 377.2 75.5 377.2 77 377.0 79 99 372.0 372.0 101.5 121 367.9 367.9 122 123.5 368.3 367.9 126.5 166.5 359.2 168 359.4 359.5 172 359.4 181 209 356.9 356.9 210 356.7 219.5 356.8 231 296.5 349.9 307 351.3 90 372.0 95.0 372.3 99.5 371.7 359.5 214
"17 bulb W
2.0 1.8 m
3.0 m
3.0 2.0 2.4 03
4 ,0 3 .0 2.8 m
4.1 3.2 3.0 m
4.0 m
1.4 1.4 2.0
co
1.0 m
1.4 1.2 m
4.0 3.1 m
3.2 m
3.4
50.7% (CHJzN2, 49.370 No 2 0 0 ~ bulb ~. 287 374.8 44.5 W 286 375.1 45.5 4.1 285 375.2 49 2.0 284 374.8 53 2.0 279 361.3 90 m 91 4.1 283 361.0 361.4 93 3.6 282 95 3.6 281 361.3 32.5% He, 180 179 177 176 186 185 172 171 170 169 174 175
GASEOUS THERMAL EXPLOSIONS
67.5% (CH&Ns 200-cc. bulb 376.8 38 m 376.8 38.5 2.0 377.0 41.5
