Bebernes, J. W.

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 5Jerrold W. BebernesIgnition for a gaseous thermal reactionIn: Michal Greguš (ed.): Equadiff 5, Proceedings of the Fifth Czechoslovak Conferenceon Differential Equations and Their Applications held in Bratislava, August 24-28, 1981.BSB B.G. Teubner Verlagsgesellschaft, Leipzig, 1982. Teubner-Texte zur Mathematik,Bd. 47. pp. 34--36.Persistent URL: http://dml.cz/dmlcz/702253Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1982Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czIGNITION FOR A GASEOUS THERMAL REACTION J. W. Bebernes Boulder, Colorado USA 1. The ignition model for a reactive gas. The ignition period of a thermal event for a perfect compressible reactive gas as derived by D. Kassoy and J. Poland [3] can be described by the following model (D t - лe - вe + x-z-ł --£--. Ja t(y,t)<iy (2) (x.O) - n(x) .x € П 5 (x.t) - 0 , x Є Э П . <= > 0 where Q C B is a bounded open container, G(x,t) i s the temperature pertur­bation of the gas, A i s the Frank-Kamenetski parameter, <y 2fc 1 i s the gas constant, and 9 (x) Is the init ial temperature perturbation. For a reactive gas in a bounded container, the associated thermal event can be violent or mild. In the former case when explosion occurs, the event is supercritical or explosive. In the latter case when there Is no dramatic event, the reactive event i s subcrltlcal. The questions we wish to answer are the following. 1) Can we describe the time-history of e(x,t) T 2) Can we distinguish between explosive and nonexploslve events as the parameters A and Y vary ? 2. Results for a solid fuel. For a solid fuel In a bounded container, the above questions were answered in [ 1 ] . Formally when -y • 1 IBVP (l)-(2) reduces to the classical ignition model for a solid fuel. (3) et - A9 - 6e* (2) e(x,o) - o , x e n » e(x,t) - o . x ^ a o , t > o The associated steady state problem Is: (4) - &f - 6e* , x € 0 (5) #(x) - 0 , x € o Q We now summarize the results for IBVP (3)-(2) and BVP (4)-(5). A] Given any bounded domain Q C K , there exists A--r > 0 such that: a) for 0 < A < 6— , BVP (4)-(5) has at least one positive solution, and b) for A > ft__ , no solution exists. The classical definition of crlticallty is based on this number A—- • Another question arises. What does A*,- nave to do with IBVP (3)-(2) • B] For any A > 0 , IBVP (3)-(2) has a unique solution e(x,t) on Q x [0,t*) , t* > 1/A , with 0 * e(x,t) sflnd-ot)"1 , 34 (x,t)€ Q x [0, 1/6). C] For any 66(0,6--!-] , IBVP (3)-(2) has a unique solution e(x,t) on n x [0,OD) with 0 « e(x,t) * 5(x) where 8 is the minimal solution of BVP (4)-(5). D] For 6 > 6-^ » IBVP (3)-(2) has a unique solution e(x,t) on (T x [0,t*) , 1/6 * t * • » wlth •UP- e(x,t) ̂  as x€n t -» t_ • Thus for 6 > 6™- the temperature 6 becomes unbounded as t -t» t— and ^ if thermal runaway occurs at t . Can we determine t ? E] Let e be the solution of IBVP (3)-(2) on Q x [o,T) . Let f(t) satisfy (6) 1 - 6e* - Xx f • f(0) - 0 on [0,T) where Xj is the first eigenvalue of: (7) - Af - Xf » x € 0 (8) f(x) - 0 , x e d Q Then sup Q(x,t) 2 $(t) on [0,T) . F] The solution s(t) of IVP(6) exists x^O on [0,T) for any 6 > 0 where (9) T-J dг ° 8e8 - Xl-T < • tf and only if 6 > Xl/e 2 6* , and lim «(t) - +• if 6 > 6* . t*T--v 6] Let 6 > 6 • Then the solution 6 (x,t) of IBVP (3)-{2) exists on 0 x [0,t*) with (10) l / 6 ^ t * | T s •• dg < * and llm^ [sup e(x,t)] - + • . Thus, for 6 > 6 a - j > 6 F R , the t+t-% thermal event for a solid fuel Is explosive since e(x,t) becomes unbounded at t and t can be estimated by the Inequality (10). 3. Results for a reactive gas. The Implicit integro-partial differential equation (1) is certainly more complicated than the Ignition model (3) for a solid fuel. This complication is due to the gas motion caused by the phenomena of thermal expansion which leads to the Integral term involving the time deviative of the temperature perturbation. The results of this section are joint with A. Bressan and will appear In detail in [2]. The first step in dealing with IBVP (l)-(2) is to put the problem in a more tractable equivalent form. Theorem 1. IBVP (l)-(2) Is equivalent to IBVP (ll)-(2) and to IBVP (12)-(2) where (11) efc - *e - 6«9 + v^Hjln1*9 + ***l *7 CM e t M a e + ^ ^ | l d o i - 6 e9 + ^ L 6 j ^ d y 35 where M i s the exterior normal to ;-, Q and dCT Is the element of surface area on ^ o • The equivalent forms follow by integrating (1) over Q to get (11) . (12) fol lows by the divergence theorem. The next two theorems are obtained using semigroup theory and invariance re su l t s for an abstract perturbation of a l inear problem associated with the l inear part of (12) . Theorem 2 . For any 6 > 0 , y k 1 , and any e o € L2(ft), *g? 0O*X* < " ' IBVP ( l ) - ( 2 ) has a unique solut ion e ( x , t ) on o x [0 f ( J) , a > 0 , where e i t h e r a < -K» and lim[sup e ( x , t ) ] « + » . Let 0 - B s {x: H x || < 1} c » n and se t BQ(x) w 0 . Theorem 3. For 6 > 0 , y * 1 , the solution 9(x,t) of IBVP (l)-(2) is non-negative, radially symmetric, and nondecreaslng on [0,<j) • As for a solid fuel, we can mathematically distinguish between explosive and non-explosive events by considering the following comparison equations: (13) q>fc - A*- 6# + $=fa 6fe*dy 0. <3) x t " i - X * oev (4) - A x - fie* Theorem 4. For 5 > 0 , y a 1 , the solution e(x,t) of IBVP (l)-(2) satisfies X(x»t>) < 6(x,t) < <p(x,t) for all x€ 0 and all t 2 0 on their common interval of existence where y is the solution of IBVP (3)-(2) and cp is the solution of IBVP (13)-(2). Since for B c lP , 6 > 6* - Xl/e > 6 ^ . the solution X of IBVP (3)-(2) •& ic fe blows up In finite time t , ve have that O < t for 6 > 6 and 9(x,t) blows up as t -* or. . Physically, this means that the temperature for an ideal gas is always greater than that for a solid in identical bounded containers and hence a gas explodes sooner than a solid fuel. Finally, ve have Theorem 5. If X is any solution of BVP (4)-(5) then £(x,t) < X(x) on Q x [0,e*) where 6 Is the solution of IBVP (l)-(2). References [1] J. Bebemet, and D. Kassoy, A Mathematical Analysis of blowup for thermal reactions - the spatially nonhomogeneous case, SIAM J. Appl. Math. 40(1981), 476-484. [2] J. Bebernes and A. Bressan, Thermal Behavior for a confined reactive gas, J. Differential Equations, to appear. [3] D. Kassoy and J. Poland, The Thermal explosion in a confined reactive gas, I-the induction period solution, submitted* 36 

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Ignition for a gaseous thermal reaction

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Ignition for a gaseous thermal reaction

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Institute of Mathematics AS CR, v. v. i.

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