T h e reaction paths are shown in Figure 1; small vertical lines on the paths mark the times 0.5 and 1.0, respectively, and circles show the equilibrium compositions. T h e heavy line represents the reaction path for the temperature profile T = (500 20 t 30 t2) ’ K., where the temperature increases continuously from 500’ K. a t t = 0 to 550’ K. a t t = 1.0. T h e nonisothermal path does not leave the envelope of the isothermal paths in this case. Figure 2 shows the dependence of mole fractions x A and xB on time for the nonisothermal case. Figure 3 gives the dependence of rate coefficients on time.
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Ac knowledgrnent
T h e authors thank Barbara Williams for performing the digital computation. literature Cited
Amundson, N. R., “Mathematical Methods in Chemical Engineering,” p. 199, Prentice Hall, Englewood Cliffs, N. J., 1966. Bilous, O., Amundson, N. R., Chem. Eng. Sci. 5,81, 115 (1956) Froment G. F., Bischoff, K. B., Chem. Eng. Sci. 17,105 (1962). Jacobson, N., “Lie Algebras,” Interscience, New York, 1962. Mah, R. S. H., Aris, R., Chem. Eng. Sci. 19, 541 (1964). Sargent, R. W. H., Trans. Inst. Chem. Engrs. 41, 51 (1963). Wei, J., Norman, E., J . Math. Phys. 4, 575 (1963). Wei, J., Prater, C. D., Aduan. Catalysis 13,203 (1962). Figure 3. Time dependence of rate coefficients ki for T = (500 20 f 30 t 2 ) O K .
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RECEIVED for review March 23, 1967 ACCEPTEDOctober 6, 1967
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D E A S T E R PROPAGATION Propagation of Exj>losiue-like Violence A.
H. M A S S 0 A N D D . F. R U D D
Department of Chemical Engineering, The University of Wisconsin, Madison, Wis. 53706 The spread of violence through weakly connected systems by a probabilistic mechanism was studied. Probability theory and simulation are used to develop preliminary criteria for the safe design of storage systems through which disaster may spread by random series of detonations. Parametric results are qualitatively compared to tlhose given by data suggested for the specification of explosives storage. This is an initial report on studiies in disaster propagation.
occasional destruction of a major processing complex to the need for further investigation into the anatomy of disaster. i4pparently, the inherent instability of certain systems to the more violent modes of operation is not known until after the system has been racked by violence. Investigations in disaster propagation should yield recommendations for the design of safer systems. Tracing the path of fire and explosion through a system indicates that disasters often consist of complex chains of events, each event triggering one or more other events. After the fact, it is often obvious how a slight change in the design of a system might have broken a critical chain of events and prevented the sweep of disaster through the system. I t is also obvious that the element of chance enters into disaster HE
T points
propagation, and that the violence moves through the system by mechanisms foreign to the common principles of transport. Some deterministic chains of events are easi!y detected and broken by the competent engineer. For example, the engineer may inspect the design of a system and observe that the chance failure of a pump will lead necessarily to a sequence of events involving the overflow of a storage tank, the flow of flammable liquid into a n area of ignition, and the final explosion of a processing component which will be surrounded by the burning liquid. This chain of events, while triggered by the chance failure of the pump, is completely deterministic in the sense that each event in the chain will occur with certainty once the chain is initiated. Often a slight modification in the design will break the chain and improve the inherent safety of the VOL 7
NO. 1
FEBRUARY 1968
131
system. This kind of a n analysis relies on the skill of the responsible engineer who should not only be familiar with the technology of the system, but also alert to the potential hazards. There has evolved a considerable body of literature in this area (Rudd and Watson, 1968). O n the other hand, stochastic chains of events constitute a major factor in the ability of a system to participate in disaster; unfortunately, these are not easily detected and evaluated. For example, the explosion of an air compressor may or may not cause the detonation of a tank of molten ammonium nitrate, which in turn may or may not cause the rupture of a hydrogen gas holder, which may or may not result in a fire which stimulates further violence. A realistic assessment of the importance of such stochastic chains of events is of great importance in process engineering, and the assessment is extremely difficult and evasive by the very nature of the phenomenon under study. Even in situations where the stochastic mechanism of the propagation of violence is fairly well defined, combinatorial problems arise to compound the chance nature of the problem to render the analysis difficult, if not impossible. For example, the blast effect following some explosions varies directly with the cube root of the amount of material involved in the explosion, and inversely with the square of the distance from the explosion site. Furthermore, the sensitivity of a neighboring storage site to detonation is a function of the blast effect it experiences. Thus, the probability of the propagation of explosive violence from site to site in a system can be defined; nevertheless, it is extremely difficult to assess the relative safety of several different ways of placing a variety of materials in a given storage area. Without such an assessment, the material may inadvertently be arranged in a decidedly unsafe configuration, a fact not known until after a major disaster. T h e propagation of violence through systems is discussed here with the hope that this will aid in exposing any potentially dangerous features. Clearly, a complete method of analysis to test all features of disaster cannot be devised. The phenomena involved are too diverse and ill defined for that. Rather, attention is confined only to certain aspects of disaster which arise primarily from systemic interactions; these are effects which arise from the combinatorial interaction of potentially hazardous components in large systems. Furthermore, in the initial phases, this work is limited to weakly connected systems in which violence is propagated from component to component by some probabilistic mechanism. T h e vehicle for this investigation is the theory of probability and Monte Carlo simulation. This report is limited to the study of disasters which develop by a n explosive-like mechanism in which the propagation can be traced as sequences of single events. A large fire which persists over a long period of time, initiating other events which propagate simultaneously through the system, would not fall within the limited scope of this first report.
tion distance defined by Robinson (1944) as the maximum separation a t which it is still possible to observe a sympathetic detonation. Robinson’s measure is approximately represented by the expression
-W -- b 7 la
where W is the weight of the exploding charge, r 2is the limiting distance, b is a constant for a given explosive, and a is a constant. Kinney, on the other hand, observes that his mean detonation distance corresponds “reasonably well” to the radius of a crater formed in a surface explosion, and uses crater size given by Robinson to construct the statement that -TmWli3
-
1.0 f 0.5
where W is the explosive energy released expressed as pounds of T N T , and yrn is the (radial) distance in feet a t which there is a “50-50 chance” of causing a sympathetic detonation. To extend the above, the probability of sympathetic detonation is greatly reduced for separation distances greater than Y,,. Figure 1, constructed from results obtained by Kinney, illustrates this probability decay, using a Gaussian ordinate. T h e probability of sympathetic detonation is expressed as a function of a parameter y which measures a separation excess as shown in Equation 3:
D
=
W1i3 (1.0 zk 0 . 5 ~ )
(3)
Here, D is the actual distance between two sites. Current explosives-handling practice employs required “magazine separation distances” giving “minimum” safe separation distances for various maximum site inventories. Typical data are summarized in Table I with the associated values for parameter y calculated from Equation 3 by equating
Sympathetic Detonation
Facilities for storing explosives must be designed with full awareness of the possibility that the explosion of one store of explosives may cause a chain of detonations that will propagate through other sites. Considering any two sites in this chain, the distance over which this propagation effect can occur is the sympathetic detonation distance. Kinney (1962) defines a mean detonation distance a t which there is a probability of 0.5 that the detonation of the second site may be caused. Reference is also made to other measures such as the limiting detona132
I&EC FUNDAMENTALS
O.OOOl:,
1
I
I
I 2 3 Excess Separation, Y
I 4
Figure 1. Sympathetic detonation probability reduction b y increased separation distance
Table 1.
Minimum Separation for Explosives StorageD
I..v, Lb. TNT
P
(E: 3 ) (Fig. 7 ) 5.42 10-6 10 8 5.36 10-6 20 10 50 14 5.60 10-0 4.90 10-5 100 16 500 29 5.32 10-6 1,000 36 5.20 10-6 5,000 61 5.16 10-6 10,000 78 5.26 10-6 20,000 98 5.24 10-6 50,000 135 6.44 10-9 100,000 185 5.95 10-9 285 7.74 10-12 200,000 385 9.54 10 -11 300,000* a Abridged from Cook (1’958).* Maximum amount permitted in any one location. 11, Ft.
where B is the blast effect, Q is the amount of material exploding, R is the distance from the explosion site, and K is a constant for a given kind of material. Furthermore, pressuresensitive material frequently detonates with a probability p = B/B* when subjected to a blast effect B, and where B* is the blast effect required for certainty of detonation. From this information one can construct the following probability statement for the propagation of an explosion from one site to another by this blast effect mechanism:
otherwise This suggests the following general form for the propagation probability for explosive-like violence :
D to the proposed minimum separation distance, and IV to the maximum site load. Olrders of magnitude for the sympathetic detonation probability are also included to indicate the approximate level of protection afforded. In general, minimum separation distances quoted by Cook (1958) are fitted by Equation 4.
D
=
3.25
5
LV”Ja6,
6
IV
< 300,000 lb. T N T
(4)
T h e above is a brief summary of typical design information that is useful, although imperfect. ‘There is considerable uncertainty inherent in statements such as Equation 2, and this reflects the lack of precise information on damage limits. Furthermore, the criterion of Equation 4 guards against loss by sympathetic detonation alone, and presumes unconstrained spacr availability. In what follo\vs, the limitation of space is incorporated in parametric analyses developed on the basis of postulates which are later shown to be compatible with current practice. Propagation Probability
T h e propagation probability is conditional in nature, and is defined as the probability that a second component will participate in disaster, given that a first component has participated. T h e initiating mechanism may be a scattering of debris following a n explosion, or the dispersion of a shock wave which interacts with pressure-sensitive material, or any combination of similar phmomena. For example, in one well documented disaster, an explosion scattered parts of vessels over a n area one-hall mile in diameter, propagating the disaster into a neighboring petroleum storage complex. I n general, the propagation probabilities are not well known and it is only for certain specific kinds of disaster that definite numbers are forthcoming. This lack of numerical value does not limit the usefulness of a parametric study in which generalizations can be drawn from trends observed in a number of reasonable cases. I n parametric studies it is necessary only to have a n accurate form for the propagation probabilities. For this form, the literature on the effects of explosive violence is scanned (Cook, 1958; Glasstone, 1957 ; Kinnry, 1962 ; Robinson, 1944). Information is available on the dispersion of material and energy following explosions. For example, from low energy chemical explosions through high energy thermonuclear explosions, dispersion of the blast effect, B, maximum overpressure in the shock wave, is approximately described by the expression
p
=
L
p’ = K’Qa/Rb; T < p ‘ 0; p’< T
