a thermal explosion model

li NOR2H- HOLLAND

A T h e r m a l E x p l o s i o n M o d e l

C. Y. Chan

Department of Mathematics University of Southwestern Louisiana Lafayette, Louisiana 70504-1010

and

P. C. Kong

Department of Mathematics Carson-Newman College Jefferson City, Tennessee 37760

Transmitted by John Casti

ABSTRACT

In a thermal explosive reaction, the rate of reaction varies rapidly with temperature and the moment of explosion occurs at some finite temperature yd. The reaction rate usually follows the Arrhenius law over a certain range of temperature [v0, Vl] with Vl ~<: Yd. A quenching model is discussed here to account for the thermal explosion. This model gives a better approximation than the blow-up model for the Arrhenius law over the range of temperature Iv0, Vl] and describes the rate of the thermal explosive reaction beyond Vl to yd.

1. INTRODUCTION

An explosion describes the spontaneous development of the rapid rate of heat release by a chemical reaction in an initially nearly homogeneous sys- tem. The rate of reaction changes rapidly with temperature. Hence, temperature may be used to describe the changes within an explosion process.

Let T < c~, and vi be a positive constant. For a simple one-dimensional explosion model with a constant density p, a constant specific heat c, and a constant thermal conductivity k, its temperature v is given by

k O2V Ov - cp- = - q ÷ + q_

v(y, O) = vi and

in (0, 1) x (0, T),

v(0, t) = v~ = v(1, t),

APPLIED MATHEMATICS AND COMPUTATION 71:201-210 (1995) (~) Elsevier Science Inc., 1995 0096-3003/95/$9.50 655 Avenue of the Americas, New York, NY 10010 SSDI 0096-3003(94)00154-V

202 c.Y. CHAN AND P. C. KONG

where y is the spatial variable, t is the time, q+ is the rate of heat release (or the rate of reaction), and q_ is the rate of heat transfer. Let us ignore q_, and assume that the rate of heat release follows the Arrhenius law, namely,

q+ = hvC~e -E/(Rv) , (1.1)

where hv '~ represents the collision frequency with constants h (> 0) and such that 0 _< a _< 1, E is the constant activation energy, and R is the

universal gas constant having the value 1.987 calories per mole per degree Kelvin. For a given chemical change, h, c~, and E are neither functions of the concentration nor of the temperature (cf. [1]). Let a = V/ -~ /k , and x -- ay. Using (1.1), we have the following initial-boundary value problem:

H v = -bv '~e -E/(Rv) in ~2, v = vi on 0g/, (1.2)

where H v = v~x - v t , b = h / (cp), gt = (0, a) × (0, T ) , and 0f~ is the parabolic boundary ([0, a] × {0})U ({0, a} >( (0, T)).

2. QUENCHING MODEL

Although the Arrhenius law is found to fit many of the available experimental kinetic data, Kuo [1] pointed out that it cannot describe the combustion process over a wide range of temperatures. It can properly describe the rate of an explosive reaction only over a certain applicable temperature range [v0, Vl] way before an explosion actually takes place.

Let us denote the right-hand side of the partial differential equation in (1.2) by - A ( v ) . D. A. Frank-Kamenetskii gave

B ( v ) = bv°L e -E / (Rv f ) e [E/(Rv~)](v-vs)

as an approximation for A(v) . This is obtained by expanding the expo- nential part in A(v ) as a Taylor series about a reference temperature v.f and taking the linear approximation. The approximation B ( v ) , called the Frank-Kamenetskii transformation, is widely used throughout the modern combustion theory, and v.f is usually chosen to be the initial temperature of the reacting mixture (cf. [2]). The problem (1.2) is approximated by the following problem,

H v = - B ( v ) in i], v -- vi on 0R. (2.1)

A characteristic of combustion is that the activation energy is very large. Hence, the assumption that the activation energy E tends to infinity in an explosion model has been made (cf. [3, 4]).

A Thermal Explosion Model 203

Let

Rv~ v = v~ + ---g-C,

bv a Ee[E / ( tlv~ )l(v~- 2v I) 6 =

Then, the problem (2.1) is described by the following initial-boundary value problem:

H e = -Se ~ in f~, ¢ = 0 on 0f~. (2.2)

Here, ¢ is usually called the nondimensional temperature (cf. [3]). Since the problem (2.2) with a = 0 occurs in the study of blow-up phenomena (namely, ¢ becomes infinite as t tends to t* for some finite time t*), we call B(v) the blow-up approximation for A(v).

We remark that this blow-up model (2.2) does not quite adequately describe the phenomenon of a thermal explosion because

(i) although the activation energy E is found to be very large in most combustion reactions, it is still finite over a given range of temperature, and hence ¢ cannot become infinite in a finite time;

(ii) the explosion of a reaction (in particular, an explosive) occurs in reality at some finite temperature, and hence ¢ cannot be infinite.

An explosive reaction is characterized by the existence of a temperature at which the reaction rate changes very suddenly or almost discontinuously. Considine and Considine [5] pointed out that although the reported "explosion temperatures" for useful "low" explosives are in the range from about 150°C to 350°C, an explosion does not actually occur at these low temperatures and requires a minimum temperature of perhaps more than 1000°C; however, this temperature gap is bridged so suddenly that the temperature difference is seldom observed by conventional experimental methods. For a "high" explosive, it may explode essentially as in the process of detonation, and the detonation temperature of such an explosive may range from as low as about 1500°C to 5500°C or higher depending on the nature of the explosive.

Table 1 gives the kinetic data for the following explosives (cf. [6]): pentaerythritol tetranitrate (PETN), 2,4,6-trinitrophenylmethylnitramine (Tetryl), cyclotrimethylenetrinitramine (RDX), and cyclotetramethylenete- tranitramine (HMX). From the data, we note that the detonation temperatures are, in fact, nmch larger than the applicable temperature ranges.

To account for the sudden rapid reaction rate at explosion or detonation, let us modify the Arrhenius law so that the resulting model gives a

204 C. Y. CHAN AND P. C. KONG

TABLE l KINETIC DATA OF EXPLOSIVES

PETN Tetryl RDX HMX

E in calories/mole 47100 38500 47100 52800 Applicable temperature

range in degrees Kelvin 433-498 483-533 488-573 543-568 Detonation temperature

in degrees Kelvin 3400 2915 2590 2365

be t t e r app rox ima t ion t han the blow-up model over the appl icable t emper - a ture range [v0, vii, and ex t rapo la tes to (Vl, Vd) where Vd is the de tona t ion t e m p e r a t u r e or the highest t e m p e r a t u r e reached dur ing the comple te combustion. Let us rewri te A(v) as

bvae-E/(Rvf)e (E/R)(1/vI-1/v), (2.3)

where v I is some reference t e m p e r a t u r e such tha t Vo <_ v I <_ Vl. Since

e ~ = l im (1 - x/p) -p, p---* oo

(1 - x/p1) -pl > (1 - x/p2) -p2 > e z for x < Pl -< P2,

it follows t h a t by choosing p sufficiently large, the expression (2.3) is well a p p r o x i m a t e d by

E 1

Let us choose p as

1) and denote the result ing Qp(v) by Q(v). Then

Q(v) = bv ~e-E/(Rw> [ v(vd - vy) ] (E/R)Cl/v,-1/,,) [vAvd - v)J

Since v$ E [vo, vii and E is very large for a combus t ion reaction, we have E / R >> v I (cf. [2]). F rom Vd >> v I, it follows tha t p is sufficiently large, and Q(v) is a good approx imat ion for A(v) in the t e m p e r a t u r e range Iv0, Vl]. Since limv--.vd Q(v) ~- o0, we call Q(v) the quenching approx ima t ion for A(v).


The following result shows that Q(v) is a bet ter approximation for A(v ) than B ( v ) over the temperature range [v0, vl], for which the Arrhenius law holds.

THEOREM. I f Vd >_ 2Vl, then B ( v ) > Q(v) > A(v ) for v • [v0,vl].

PROOF. By construction, Q(v) > A(v) . Let

p B ( v ) - v - v : , vy

P Q ( v ) = Vd- -V : In lV(Vd---VI)[ Vd iv: (v~ - v)J"

Then,

B ( v ) - Q ( v ) = by% -~ / (Rv~) (e [~/(R's)IP"(') - e [~ / (R's ) leQ( ' ) ) .

To show that B ( v ) >__ Q(v) for v • [vo,vl], it is sufficient to show that PB(v) >_ PQ(v) for v • Iv0, vl]. Let

D(v) = PB(V) -- PQ(v).

Then, D ( v f ) = 0 and

dD (v - v:)(Vd -- V: -- V)

dv v:v(vd - v)

For critical values, we have d D / d v = 0 which gives v = v: or v = Vd -- v I. Since Vd >_ 2vl, it follows that

dD

dv dD

- - < 0 for v • (0, v : ) u (v~ - v : , vd),

d--~ > 0 for v • (v f , Vd -- v: ) .

Therefore, v = v I is a minimum point and v = Vd - -v I is a maximum point. Because D ( v : ) = 0 and v = vi is a minimum point, we have D(v) > 0 for 0 < v < Vd -- v: . Since vd _> 2vl and v/ < Vl, it follows that

PB(V) >_ PQ(v) for v • [Vo,Vl].

Hence,

B ( v ) > Q ( v ) for v e [vo, vi i . •

206 C. Y. CHAN AND P. C. KONG

TABLE 2

COMPARISON OF ErrAn_B AND ErrAn_Q

Err_in_B

v d oL vf Err_in_B ErrAn_Q Err_in_Q

0.0 489.92 1.67 × 10 -4 5.51 x 10 -5 3.02 PETN 3400 0.5 490.00 1.60 × 10 -4 5.30 x 10 -5 3.02

1.0 490.08 1.54 x 10 -4 5.10 x 10 -5 3.02

0.0 522.17 2.15 x 10 -4 7.83 x 10 -5 2.75 Tetryl 2915 0.5 522.28 2.07 x 10 -4 7.53 x 10 -5 2.75

1.0 522.41 1.99 × 10 -4 7.25 x 10 -5 2.75

0.0 562.35 2.18 x 10 -4 8.61 x 10 -5 2.54 RDX 2590 0.5 562.47 2.09 x i0 -4 8.23 x 10 -5 2.54

1.0 562.56 1.99 x 10 -4 7.87 x 10 -5 2.53

0.0 560.31 4.97 x 10 -5 2.12 x 10 -5 2.34 HMX 2365 0.5 560.36 4.89 x 10 -5 2.09 x 10 -5 2.34

1.0 560.40 4.82 × 10 -5 2.06 x 10 -5 2.34

For n u m e r i c a l i l l u s t r a t ions , le t us c o m p a r e s o m e n u m e r i c a l d a t a be -

t w e e n t h e two a p p r o x i m a t i o n s B(v) a n d Q(v). For any g iven [co, Vl] a n d

E , we choose t h e re fe rence t e m p e r a t u r e v I E [co, Vl] such t h a t for v E

[vo,vl],B(v) gives t h e bes t a p p r o x i m a t i o n for A(v) by us ing t h e leas t

squa re s m e t h o d . Le t

Er r_ in_B = m i n f~2 (A(v) - B(v) ) 2 dv v0<v,<v, f~2(A(v))2dv

For c o m p a r i s o n , we use t h e s a m e re fe rence t e m p e r a t u r e v I to c o m p u t e t h e

r e l a t i ve e r ro r in us ing t h e q u e n c h i n g a p p r o x i m a t i o n Q(v) by

E r r A n _ Q = f~l (A(v) - Q ( v ) ) 2 dv f~o ~ ( A ( v ) ) 2 dv

F r o m T a b l e 1, we n o t e Vd > 2Vl. U s i n g t h e d a t a the re , we o b t a i n t h e n u m e r i c a l r e su l t s in T a b l e 2 by M a t h c a d ve r s ion 4.0 (w i th t o l e r a n c e -~

0.0001). T h e n u m e r i c a l r esu l t s show t h a t Q(v) is a b e t t e r a p p r o x i m a t i o n

for A(v) over t h e t e m p e r a t u r e r a n g e [vo, vl] . W e n o t e t h a t t h e l a rge r t h e

d e t o n a t i o n t e m p e r a t u r e Vd, t h e b e t t e r t h e a p p r o x i m a t i o n Q(v) as c o m p a r e d

w i t h B(v) . Since t h e va lue o f t h e d e t o n a t i o n t e m p e r a t u r e is a f fec ted g r e a t l y by con-

d i t ions , e spec i a l l y c h a r g e d e n s i t y a n d p re s su re (cf. [7]), we w o u l d like t o see

t h e v a r i a t i o n of Err_ in_Q w i t h Yd. For t h e c h e m i c a l exp los ive P E T N w i t h

v f = 489.92 K a n d ~ --- 0, we o b t a i n t h e n u m e r i c a l d a t a l i s ted in T a b l e 3


TABLE 3 EFFECT OF DIFFERENT v d

vd Err_in_Q Err_in_B/Err.in_Q

3400 5.51 x 10 -5 3.02 34000 4.15 x 10 -5 4.02

340000 4.04 x 10 -5 4.12 3400000 4.03 x 10 -5 4.13

by using Mathcad. The numerical results indicate that increasing Vd does not reduce Err_in_Q much. Indeed, in the quenching approximation Q(v),

P = ~

When Vd is larger, p is larger and this in turn gives a smaller value of Err_in_Q. In fact, p is bounded by E/(Rvo) as Vd tends to infinity. In other words, the quenching model is not much affected by the inaccurate information on yd.

We understand that there are, in fact, many ways of improving an approximation for the Arrhenius law A(v). For example, we know that

B(v) = bv~ e-E/(Rvl) e[E/(Rv~ )](v-vD,

and (1 + x/q) q < e ~ for any q > 0. If we let

B q ( v ) = 1 + ( v - ,

then for a suitable choice of q, we can have B(v) >_ Bq(v) ~- A(v) for v E [v0,vl]. Though Bq(v) can be a close approximation for A(v) for v E [vo,vl], it cannot explain the fact that an explosion takes place at some finite temperature. On the other hand, Q(v) can be refined to give a better approximation for A(v). One way of doing this is as follows:

Since (1 + x /n ) '~ <_ e x <_ (1 - x / n ) -n for any n > x >_ 0, and

E

let

• /1 p }

208 C.Y. CHAN AND P. C. KONG

for some/3 E (0, 1). Then for a suitable choice of/3, Q~(v) can be made to almost coincide with A(v) over the temperature range for which the Arrhenius law holds. We note that Qz(v) tends to oc as v tends to Vd, which may correspond to the occurrence of a thermal explosion at a finite temperature.

Although Q~(v) is a more accurate approximation for A(v) when v E [v0, vl], it is a more complicated model because we have to choose v f and /3. To keep the model simple, a reasonably good approximation for the Arrhenius law over Iv0, vii is given by the quenching approximation Q(v), which can extend the temperature range to Iv0, Vd). Thus, the quenching model for the thermal explosion process is given by the following initial- boundary value problem,

Hv = -Q(v) in ~, v = vi on 0it.

Using the transformation u -- v - v~, we have the following thermal explosion model:

where

Hu = - f ( u ) in it, u = 0 on 0it, (2.4)

K (u + vi) a+p f(u) = (v-~: v - : : u)-~'

v! /

We note that

I'(u) = g ( u + ~)"-2MP÷I[(~ + P)~d -- ~(U + ~)1, f ' (U) ---- g ( ~ + v~) ~-~

x MP[(a + p - 1)(a + p) + 2p(a + p)M + p(p + 1)M21,

where M = ( u + vi)/(Vd- v i - u). Hence, f(O) > O, f ' > O, f " > 0 provided that p > ( 1 - a ) ( 1 - v i / v a ) 2, and lim~,--.Vd-V, f(u) = oo. In fact, the problem (2.4) is the quenching problem, which has recently been studied by many mathematicians [8-24] since its introduction in 1975 by Kawaxada [25].

REFERENCES

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7. H. Henkin and R. McGill, Rates of explosive decomposition of explosives, experimental and theoretical kinetic study as a function of temperature, Ind. Engrg. Chem. 44:1391-1395 (1952).

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9. C. Y. Chan and S. S. Cobb, Critical lengths for semilinear singular parabolic mixed boundary-value problems, Quart. Appl. Math. 49:497-506 (1991).

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11. C.Y. Chan and L. Ke, Critical lengths for periodic solutions of semilinear parabolic systems, Dynamic Systems Appl. 1:3-11 (1992).

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210 C.Y. CHAN AND P. C. KONG

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25. H. Kawarada, On solutions of initial-boundary problem for ut = ux~ + i/(1 - u ) , Publ. Res. Inst. Math. Sci. 10:729-736 (1975).

a thermal explosion model

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