Combustion and Flame 113

The Dependence of Spontaneous Ignition Temperature on Surface to Volume

Ratio in Static Systems for Fuels Showing a Negative Temperature Coefficient

B. F. Gray School of Chemistry. University of Leeds. Leeds, England

Certain hitherto unexplained features in tile observed variation of spontaneous ignition temperatures with surlhce-to- volume ratio are discussed in terms of chain-thermal ignition theory. They are shown to be expected for substances that exhibit cool flames and two-stage ignitions, as i~, a region of"two-stage instability" at higher s/v values, not yet observed. Thl~ results are relevant to lhe ass~tssmenl of harards associalud with spontaneotls ignition in enclosed compartnlcnts sttch as aircraft I'uel tanks.

I n t ro d uc t i on In a recent experimental investigation [ I ] the spontaneous ignition temperature of kerosene was measured as a function of the surface/ volume ratio. The method utilized steel inserts in a steel reaction vessel, and the results were shown to agree well with those of an earlier investigation [2"1 in which the reaction vessel diameter itself was varied, in both steel and Pyrex vessels. The results are shown in Fig. I, the main point of interest being the discontinuity in slope at about 250°C. This is not understood. but the results (curve ,4BD) in Fig. I are believed to indicate that the risk of spontaneous ignition in, tbr example, an aircraft compartment such as a fuel tank would be reduced by inserting a metal honeycomb in order to increase the sly ratio [I ].

In this paper we will attempt to show that this conclusion is not necessarily correct and hence is possibly hazardous. The discontinuity in slope in Fig. 1 is very significant, as it can be shown to arise from either a negative tempera-

UNSTABLE

. . . . . . . . F @.~" -CNSTA BL ~ C

; - a

A i

t oc (s/v)

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Published by Amerlean Elsevier Publishing Compa i~, Inc,

114

ture coefficient in the heat release rate curve or at leas~: a point of inflection, thus allowing a double tangency condition for the heat release and loss curves on the usual type of thermal diagram. Heat release curves of this type have long been known to be related to oscillatory cool flames and lobes on (P-TO) ignition diagrams. In these eases ignition occurs on decreasing TO in certain conditions, which is the opposite of what might be expected from extrapolation of measurements of normal behavior made out- side this range. The existence of cool flames in kerosene oxidation is additional evidence that there may be a negative temperature coefficient and hence "abnormal" ignition behavior. A theoretical interpretation of oscillatory cool flames, ignition lobes, and the negative tempera- ture coefficient has been given in detail recently elsewhere [3.415,6.7]. and the main results of this work will be used here without proof.

T-s / v C u r v e on Simple T h e r m a l T h e o r y The equations of simple explosion theory [8] are at the critical condition

R(T,. e ) - / ( r , - TO) = o 0)

dR/d?~- / = 0 (2)

where I ~ sly, and T~ is the critical temperature of the reacting mixture at ambient temperature TO. P is the pressure, which in this discussion will be held constant. Solving Eq, I for T~ and differentiating with respect to L we obtain

dT o .. dT¢ R(~, P) (dR/d~) (dT,./dl)

dT~ ~ dR/dT¢'~ P R - - -7- - ) + (3)

and~ using Eq. 2, we see that the first term vanishes a'. the critical condition: hence

dTo/dl = R(~, P)/I 2 = (T,. - "fro)//> 0 (4)

tt is worth noting that at this stage we have

B. F, Gray

made no assumption about the detailed form of R; yet (dTo/dl> 0 is always true provided the reaction is exothermic. If we assume a simple Arrhenius form for R, then, following Semenov, we can derive

T c- T o = RT~/E (5)

which makes Eq. 4 integrable to

/ = Ce- ~/Rro (6)

where C is an integration constant. For values of E/RTO near criticality, this gives a curve similar to the separate branches of the curve in Fig. I, that is. convex upward.

T - s / v Curve on M o r e G e n e r a l T h e r m a l

T h e o r y The discontinuity in dTo/dl observed in Fig. I. according to Eq. 4. implies a discontinuity in T~ as a function either of T O or of / . This can (and does) readily occur in systems exhibiting a negative temperature coefficient in the heat release rate (Fig. 2). When criticality occurs in the region of point A, increasing I (the slope of the straight line) will eventually, after a double tangency condition, result in criticality occurring in the region of point B, T~ having suffered a dis- continuity. Equation 4 then tells us that we would therefore expect a discontinuity such as that ob- served by White [ I ] .

F~GURE 2

R,/.

T

Spontaneous Ignition Temperature and s/v Ratio

I f this is the correct explanation of the dis- continuity in Fig. 1, consideration of the results proved in earlier work [3,4,5,6] leads to the conclusion that a region of ignition will exist in Fig. 1 at larger values of I than that at which the discondntdty arises. The essential result needed here is that, within the region of negative tem- perature coefficient, there is a region XY in Fig. 2 such that the steady state: (although existent) is not accessible from the normal initial co~,ditions of the system (i.e, when the initial temperature of the reacting gas is equal to the ambient temperature). This possibility arises as a result ofcltain branching in the system: and, very crudely speaking, when the system has attained the steady-state temperature between X and Y, the radical concentration is far in excess of its equilibrium value, and the system undergoes a two-stage ignition.

Simple geometrical arguments based on Fig. 2 show that the region of two-stage instability X Y in the heat release curve (which accounts for the lobe on a P-To diagr~,m) gives a region of instability CEFin Fig. I. bounded below by the continuation of the tangency condition at A in Fig. 2. The latter, of course, ceases to represent the critical condition at B in Fig. I. which represents the double tangency condition. Point E in Fig. 2 occurs when the tangent at A in this figure intersects the heat release curve again at Y, etc. The unstable region CEF would not

115

occur, in spite of the discontinuity in ABD. if the latter were caused simply by an inflection in the heat release curve or one with a very weak negative temperature coefficient, since then the two-st~,ge ignition region would not be present. However, as cool flames have been observed in kerosene oxidation, there must be a sizable negative temperature coefficient and hence the probability of a region of two-stage ignition such as CEF in Fig. 1 and its potentially dis- astrous consequences in the context of aircraft fuel tanks and other hazardous situations. Further experimental work is this area would seem to be desirable.

Refe rences I. WHITE. R, G.. "Spontaneous ignition Of kerosene

vapour." Royal A ircraJ~ Establishment Teeh. Rept. 67107 (May 1967).

2. KUCHTA. J. M.. BARTKOWIAK. A.. and ZABETAK[S. M. G., J. Chem. Eng. Data. 1O. 282 (196S).

3. GI~AV. B, F,. Trans. Faraday Soc.. 65. 1603 (1969). .4. YANG, C. H.. and GRAY. B. F.. Trans Faraday Sot.. 65.

1614 (1969). 5. GRAY. B. F,. and YANG. C. H,. J. PIo's, Chem.. 69. 2747

(1965). 6, VEDENEEV, V [.. GERSHI~NZON. Jg M.. and SARKISOV. O,

M,. Armenian Chem. J,. 20. 968 (1967). 7, YANG. C. H., Private communication. 8. SEMENOFF. N. N., Chemicall(ineticsaudChain Reaetio¢ts.

Pergamon : Oxford (1953).

(Received May 1969; revised June [969)