James S. Wright Corleton University Ottawa. Ontario Canada
II
Kinetic Model of a Thermal Explosion
The exact solutions to the mathematical treatment of a thermal explosion, including mass and heat transfer, were obtained by Frank-Kamenetskii ( 1 ) and Rice and coworkers (2) in the 1940's. Approximate treatments of the problem have been given hy Semenoff (3, 4) and discussed by Benson (5). In this paper we discuss the simplest possible model which can explain the existence of an induction period and a sudden rapid temperature rise. The calculation of maximum flame temnerature hv Glasstone (6) follows a similar approach, except that kinetics then play no role. Consider the following model system: The unimolecular gas phase reaction B --c occursin a closed vessel of volume V. The heat of reaction Q = AE is liberated for each mole of B reacted, and AE is assumed constant with temperature. The heat liherated is assumed to he uniformly distributed in the vessel, which is insulated, so that all heat generated goes into raising the temperatures of B and C, which have specific heats CdB) and C,(C), respectively. As the temperature rises, the reaction rate increases according to the. Arrhenius' expression k = AecEaIRT. This leads to faster production of C, faster heat evolution, and so on (i.e., heat catalyzes the reaction). The pressure, which builds up inside the vessel, is related to the internal temperature by the perfect gas law. An explosion occurs if the pressure generated before reactant B is depleted is sufficient to rupture the vessel wall. Given this model, would one expect an induction period followed by a catastrophic temperature rise, or a more gradual increase? A computer solution is ohtained in the following way: Assume a vessel volume V (I), a vessel wall strength (atm), an initial temperature To ("K), an initial amount of B present n," (mole), a heat of reaction Q = A F = AH" (kcal/mole), an activation energy E, (kcal/mole), a frequency factor A (See-'), and heat capacities CdB) and C,(C) (cal/mole deg). Calculatethe rate constant according to = Ae-E"!RT
Allowing time interval At to elapse, calculate the amount of B remainine hv - the first-order rate exnression B = Boe-hA' The amount of C produced = (Bo - B) = amount of B decomposed. The amount of heat produced in time interval At is Q = (&-B)AP The temperature rise
so the new temperature is To + A T degrees. The pressure is P = (ns +nr)RT/V = (ne")RT/V
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for the reaction B C, where there is no change in the total number of moles. If the pressure is less than that needed for an explosion and some B still remains, recalculate the rate constant k a t the new temperature and repeat the cycle (For the next cycle B0 is the amount of B left after the previous cycle, and simila~lyfor the other variables.)
J-+
1.
INDUCTION PERIOD
I0
0.5
1.0
1.5
21
2.5
TIME Issc I T e m p e r a t u r e and p r e s s u r e behavior
as a function of t i m e
In a trial run, using values similar to those which pertain to the thermal explosion of azomethane (Zc), it was assumed that T = 600"K, V = 1 1, . n ~ "= 0.02 mole, CdB) = C,(C) = 15 cal/mole deg. AE" = 43 kcal/mole, E, = 51 kcal/mole, A = 1.0 x 1016 sec-1, time interval between calculations = 0.0002 sec. The figure shows a plot of (calcd) pressure versus.time for these data. In the region 0-1.90 sec there is a gradual pressure rise from 0.984-1.08 atm, corresponding to a temperature increase of 57°K. At 1.94 sec the reaction "goes critical," the temperature rises by 2800°K in 0.02 sec, and the pressure rises to its final value of 5.69 atm. Assuming a vessel capable of withstanding 3 atm total pressure, an explosion will occur at 1.96 sec. Of the many simplifications used in this model, the most serious are the ignoring of the resulting shock wave, which could induce failure in much stronger vessels, and the assumption of an instantaneous uniform distribution of the heat generated. However, the answer to whether or not the system will experience a catastrophic temperature rise is certainly unequivocal. An interesting exercise for students is to extend this model to make it correspond more closely with reality. Possible directions could be in allowing for heat transfer through the vessel walls, allowing for changes in AE and C, with temperature, introducD tion of more complex kinetic schemes such as B + C E, and in the search for data which correspond to a real system. In the process of working through the solution, a valuable link between thermodynamics and kinetics is established.
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Literature Cited (11 Frank~Kamenefskii.D. A., "Diffuion and Hear Erehsnge in Chemical Kinetics." Princeton Univ~raityPIOBS,P r i n ~ e t 0 9NJ., 1955. IOriginsll) published in Russian in 1947) (2) la) Allen. A. 0. and Rice. 0.K.. J. Amer Chpm Soe ST. 310 119351: lhl Rice. 0. K.. Allen. A. 0.. and Camphdl. H. C.. J. Amsr Chrm. Soc.. 57. 2212 11935); Icl Rice, 0 . K . , J Chem Phys.. 8.727 119401. (31 Somenoff, N.. "Chemical Kinetics and Chain Reactions." Ciarendan Press, Oxford, 1935. (11 Semenoff. N.. "Some Prablema in Chemical Kinetics and Reactivity." Vol 2. Princeton University Press, Princeton, N.J.. 1959, p 87. (51 Benwn. S., "The Foundations of Chemical Kinetics," McGraw-Hill Book Co.. N.Y.. 1 9 6 0 . ~ 4 3 1 . (61 Glesstone. S.. "Thermodynamics for Chemists," 0. Van Nostrand. New York. 1947, p84.
Volume 5 1, Number 7, July 1974 /
457
