Structure of the thermal explosion limit

COMBUSTION A N D F L A M E 72:221-224 (1988)

221

Structure of the Thermal Explosion Limit V. I. B A B U S H O K Institute o f Chemical Kinetics and Combustion, Novosibirsk, USSR

and V. M. G O L ' D S H T E I N Institute o f Mathematics, Novosibirsk, USSR

in a closed system an explosion limit is the transition region which separates the regions of slow and explosive reactions with respect to initial conditions. In the present paper the characteristic features of chemical reaction in the transition region are investigated for thermal explosion. The critical phase trajectory of the system contains an unstable integral manifold. Initial conditions corresponding to this trajectory can be understood as the explosion limit. An equation determining maximum temperature rise on the critical trajectory is obtained. The transition region can be divided into regions of slow and explosive transient regimes. The transition region is characterized by the extrema of induction period and degree of reactant consumption which correspond to a maximum reaction rate.

INTRODUCTION An explosion limit means a set of initial conditions at which a slow reaction changes abruptly to a fast, explosive one. In a closed system the explosion limit is a narrow range of parameters where the reaction behavior changes. When reactant consumption is taken into account, the transition region presents difficulties in determining the critical conditions [1, 2]. The characteristic features of reaction behavior in the transition region are analyzed in the present paper; i.e., the structure of the explosion limit is investigated. A model of thermal explosion with reactant consumption is taken as an example. The limit structure means the regularities in the changes in the basic reaction characteristics (rate, induction period, etc.) with varying initial conditions in the transition region. The main types of transient regimes can be distinguished. The critical trajectory (unstable integral manifold) is distinCopyright © 1988 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, NY 10017

guished naturally. Conditions corresponding to this trajectory comprise the explosion limit proper in its conventional sense. To demonstrate the characteristic features of the transient reaction behavior, in contrast to the explosive and slow regimes, we consider the transition region as an independent subject of inquiry, with its own peculiar regularities. Some aspects of the reaction behavior in the transition region have been discussed when investigating critical conditions in terms of the nonstationary theory of thermal explosion [1, 2] and when analyzing thermal explosion degenerate regimes (explosion with extensive reactant consumption) [3, 4]. The explosion limit structure has been studied in Ref. [51 and [6]. RESULTS AND DISCUSSION In terms of traditional dimensionless variables a thermal explosion model (first-order reaction) has

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222

V . I . BABUSHOK and V. M. GOL'DSHTEIN

the form 3' drr = .7 " exp

Se '

d---~-=-*7 . e x p

.

(1)

Here/9 and '7 are the dimensionless temperature rise and the reactant concentration, r is the time scaled by the characteristic chemical time, Se is the Semenov number, ~ is the invcrse of the adiabatic temperature rise, and ~ is the reduced ambient temperature. The analysis is carried out on the phase plane (•, *7), Thermal explosion on a phase plane has been studied in Refs. [7] and [8]. When analyzing system (1) the small value of 3~ allows the division into rapid and slow motions [9] and the use of one of the modifications of the integral (invariant) manifold method [10]. In analyzing system (1) the above division has previously been used in Ref. [11]. The manifold (smooth surface) M lying in R m+" x [0, 3'0] is referred to as an integral (invariant) one for the system

--~=f(x, y),

"y

=g(x, y )

(2)

(0 <_ "t <- ?o, X (3. Rm, y E R"), ifin this system each trajectory having at least one common point with M lies entirely on M. A local integral manifold is defined in the same manner. We shall not distinguish terminologically the local and nonlocal cases. We are interested only in manifolds which are plots of mappings y = h(x, ?) under the condition h(x, "y) ~ ho(x), if 3' ~ 0. The mapping y = ho(x) specifies one of the branches of the slow surface g(x, y) = 0 [g(x, ho(x)) = 0]. Solutions of system (2) in the zeroth approximation O" = 0) (the so-called discontinuous solutions) include rapid and slow portions. The slow portions lie on the slow surface. Figuratively speaking, the integral manifold is a refinement of the slow portions of the discontinuous solution including the small parameter 3,; i.e., the true trajectory with its slow portion lying on the integral manifold is separated from the slow surface by a distance which is O(3'). The integral

manifold position is calculated by a formal expansion in terms of the small parameter [9]. The integral manifold can be understood as the surface of a phase rate local minimum. Note that the division of the motions on the phase plane (rapid, slow) does not necessarily correspond to the division (with respect to the same principle) of rates of change of the variables with respect to time. The slow curve F: .7 = 0 exp [ - 0/(1 + /30)]/ Se of system (1) has two stable branches and an unstable one (FI,/'2, and Fo, respectively, Fig. 1). There is a narrow bundle of trajectories which go over F0 and nearly along a substantial portion of it. The following relation between the parameters/3 and 3' is characteristic of conventional explosive gas mixture: 13 > 3' > 132. The adiabatic temperature rise equals 1/% Therefore the path having the initial point (0, 1) must cross F0 (Fig. 1). There is a limiting point (C) for the motion along F0 (Fig. 1). The coincidence of the slope of the line tangent to F0 with that of the adiabatic straight line is the criterion which determines this point. On the portions of temperature rise curves there are no trajectories of the system which follow any portion of the curve F0 between the points C and D (Fig. 1); i.e. the integral manifold close to F0 does not exist within this portion. The value of 0c is approximately determined from the equation Oc- l = 7 exp(Oc- 1).

(3)

As stated above, at a distance of 0(3,) from the slow curve unstable branch F0 there is an unstable integral manifold [9, 10] which is the plot of the

1(

le.Se)"I

o

o

1.2p

1/~

~/j~2

e

Fig. 1. Phase portrait of the system. Arrows show direction of trajectories. A denotes the slow curve maximum; B denotes the slow curve minimum. For details see text.

THERMAL EXPLOSION LIMIT function 0 = 0(71, 7). This manifold is a portion of one of the trajectories of the system. With an appropriate choice of Se this trajectory (I0) can start at the point (0, 1). It is natural to consider I 0 to be the critical trajectory and the value Seo, at which I0 starts at (0, 1), to be the explosion limit. The regime corresponding to the critical trajectory is neither slow, since its temperature rise is essentially higher than 1, nor explosive, because the temperature (from 0 --- 1) rises as fast as the slow variable changes. The critical trajectory at the portion close to F0 is repulsive (saddle type), as shown in Fig. 2. It is the longest slow path along F0. It shows the longest slow temperature rise (on the phase plane) which does not lead to explosion. The path portions almost parallel to the adiabat correspond to explosion. To analyze the transient behavior close to the critical one it is necessary to use the first approximation of the integral manifold as well as the refined ones. The first approximation has the form (1 +/300(r/))4002(r/) 01(r/, "y) = 0 0 ( r / ) Jr- "y

223

1

9

/.

0.8 0.6 0,/. 0.2

_

0

'~ 0

, 2

T-4

,~"-"--'---.~. 6 8 10

@

Fig. 2. Example of numerical calculation of reaction transient regimes on the phase plane, /3, 3' = l0 -2. The slow curve corresponds to Se = 0.4211133. Se: l, 0.39; 2, 0.42; 3, 0.4211133; 4, 0.425; 5, 0.42125; 6, 0.421113605. Other curves were calculated for the average Se = 0.4211133085 and initial conditions, respectively: 0 = 3, 4, 5, 7 and ~/ = "qF + 3', ~V + 23'; "qF = 0/(Se-exp[0/(1 + 80)]).

manifold. In the case/3 = 0 this has the form 02(71, "V)=01(r/, 3 ' ) + 7 2 " S C 00(r/) • exp(200(r/)) X

(00(rl)- 1) 5

× (6002(r/) + 30o(*/) -

2).

r/(00(r/) - (1 +/300(r/))2) 2 '

where 00(r/) is the solution of the equation for F0 with respect to 0. It can be solved with respect to r/ and written explicitly as 1'/1(0, ~ )

0(1 + / 3 0 ) 2

0 Se • exp[0/(1 +/30)]

Y

(1 +/30) 2 - 0 "

The first approximation is a portion of the curve d20/dr/2 = 0. At 0 = 2/(1 - 2/3) the curve changes the sign of its curvature. The paths which are along I0 and below it (Fig. 2) describe slow transient regimes. In the first approximation the trajectories lying below the curve d20/dr/2 = 0 correspond to them. Therefore they cannot change the sign of their curvature and at the values of 0 < 2/(1 - 2/3) they have to move away from Fo and approach rather sharply the slow curve stable branch F~. Thus, the slow regimes of the first approximation reach the temperature rise 0 < 2/(1 2/3). To study the slow transient regimes for a temperature rise more than 2/(1 - 2/3) requires the use of the second approximation to the integral

As shown in Ref. [5], asymptotic methods are applicable to investigate the critical trajectory at 1 < 0 < 0, where 0 is the sufficiently accurate solution of the equation 3,e ° I -- 1/(0 - l) and corresponds to the point E in Fig. 1. It is shown in Ref. [5] that the main conclusions are valid also at ~<0<0c. With respect to the initial conditions the transition region can be divided into slow and explosive transient regimes. The paths along and below the unstable integral manifold refer to the slow transient regimes. The trajectories along and above the unstable integral manifold correspond to the explosive transient behavior. In the region of the slow transient behavior with rising Se an increase is observed in induction period rw, maximum reaction rate Win, maximum temperature 0m and in the degree of reactant consumption 1 - r/w, corresponding to the maximum reaction rate. The minimum value of r/w is obtained at Seo which refers to the critical trajectory [r/w = (1 to 3)3']. Further increase in Se leads to " b r e a k - o f f " from the critical trajectory, which corresponds to explosive behavior in the transition region. At Se > Seo

224 the reaction proceeds by two stages. The first stage is a rather slow one up to considerable degrees of consumption (motion along F0). The second stage is an explosive one (motion parallel to the adiabat). As follows from above, explosive phenomena can be observed at r/values close to 3'. In the present scale-up the equality ~/ = 3" refers to a maximum temperature rise A0 = 1. Extrapolating these results, it probably can be stated that if a system contains more than a fraction 3' of explosive substance, then initial conditions can be found which cause explosive phenomena in the system. The ~'w extremum as a characteristic feature of the transition region has been mentioned in Refs. [1, 3-6, 12]. The analysis in Ref. [5] shows the rw extremum to refer to the critical trajectory. Note also that in the transition region there is the extremal degree of reactant consumption 1 - ~w corresponding to the maximum reaction rate. An upper limit to a quantity 0 on the critical trajectory is the critical temperature rise in the transition region. Its value is quite an accurate solution of (3). At/3, 3' = 10 -2, 0c = 7.8. The trajectories (after the induction period) along the stable branch F1 of the slow curve [7, 12] (Fig. 2) correspond to the slow stage of the reaction. For this stage the maximum temperature rise does not exceed 1 + 2/5. For the slow transient regimes a temperature rise up to 0c is possible. In the zeroth approximation the rate of temperature change in time at the critical trajectory at 1 < 0 < 0c is determined by the equation dO~dr = [O/ (0 - 1)]e °. In the region 0 = 1-2 the time dependence of the solution of this equation is approximately linear. At 0 > 2 the exponential rise becomes pronounced. As can be seen from Fig. 2, phase trajectories of motion of the system in the transition region differ from those of the slow and explosive regimes. CONCLUSION Thus, the transition region is shown to have a fairly complex structure. The sequential change in type of phase trajectories corresponds to a transition from the slow reaction regime to the explosive one. Trajectories passing along the stable branch Fj of the slow curve (slow reaction) are changed

V . I . BABUSHOK and V. M. G O L ' D S H T E I N into those along the unstable integral manifold (transient regimes) and then into explosive ones parallel to the adiabat. The critical trajectory is naturally exceptional as the one containing the unstable manifold. The present paper does not seek to calculate the Se critical value (the explosion limit) corresponding to the critical trajectory. This value can be determined by the asymptotic expansion technique.

The authors express their thanks to O. M. Todes, V. S. Babkin, and V. A. Sobolev for useful discussions and are grateful to A. S. Romanov for verifying the calculation of the integral-manifoM second approximation by the REDUCE program, as well as for obtaining relations for the third and fourth approximations. REFERENCES 1. Merzhanov, A. G., and Dubovitskii, F. I., Uspekhi Khimii 35:656-683 (1966) (in Russian). 2. Gray. P., and Sherrington, M. E., in Gas Kinetics and Energy Transfer. A Specialist Periodical Reports, The Chem. Soc., London, 1977, Vol. 2, pp. 331-383. 3. Merzhanov, A. G,, Zelikman, E. G., and Abramov, V, G., Dokl. Akad. Nauk SSSR 180:639-642 (1968) (in Russian). 4. Boddington, T., Gray, P., Kordylewski, W., and Scott, S. K., Proc. R. Soc. London A390:13-30 (1983). 5. Babushok, V. I., and Gol'dshtein, V. M., Explosion Limit: Transient Regimes o f Reaction, Institute of Mathematics, Novosibirsk, 1985, prepr. No 10, pp. 3-51 (in Russian). 6. Babushok, V. I., Krakhtinova, T. V., and Babkin, V. S., Kinetics and Catalysis (transl. Kinetika i Kataliz) 25:5 (1984). 7. Todes, O. M., and Melentyev, P. V., Z. Fiz. Khimii 13:1594-1609 (1939) (in Russian). 8. Adler, G., and Enig, J. W., Combust. Flame 8:97-103 (1964). 9. Mishchenko, E. F., and Rozov, N. K., Differential Equations with Small Parametem and Relaxation (transl. from Russian by F. M. C. Goodspeed), Plenum Press, New York and London, 1980. 10. Bogolyubov, N. N., and Mitropol'skii, Yu. A., Asymptotic Methods in the Theory o f Nonlinear Oscillations, Gordon and Breach, 1967. 11. Gray, B. F., Combust. Flame 21:317-325 (1973). 12. Kassoy, D. R., and Lift,in, A., Qrt. d. Mech. Appl. Math. 31:99-111 (1978). Received 7 April 1987; revised 11 August 1987