THEORY OF THE SURFACE IGNITION ENERGY
OF CONDENSED EXPLOSIVES W .
H . ANDERSEN
Shock Hydrodynamics. Inc., Sherman Oaks, Calif
A simple theory that elucidates the important featu.res of the surface ignition energy and delay time of an explosive under high heat flux i s developed from the heat conduction and thermal decomposition kinetic processes involved. The ignition energies predicted for various explosives are in agreement with those measured b y Bryan and Noonan. The ignition temperatures predicted for these explosives are consistent with those published in the literature. The theory predicts an increase in the required ignition energy with increasing ignition delay time as was found b y Jones. geometry of the heat transfer process.
THE
ONE-DIMESSIONAL surface ignition of a n explosive from a pure thermal heat source with defined heating characteristics is knoivn from experiment to require a certain critical energy per unit area from the heat source, below which ignition will not occur though the explosive adjacent to the source may be induced to undergo some chemical reaction. Thus, a selfsustained combustion-i.e., ignition-can result only when the rate of heat production by the chemical reaction is greater than the rate of heat loss to the surroundings. As to whether the ignition results in normal combustion, or a thermal explosion followed by detonation, is a matter ivhich 'is related to the kinetic nature of the decomposition of the explosive, the ambient pressure, and the physical conditions-e.g., packing density, particle size. charge diameter, confinement-of the explosive. This topic is beyond the scope of this paper. Experimental values of the thermal energy required to ignite an explosive are meager. Morgan ( 9 ) found that the energy required for the ignition of highly flammable solids increases with the time during which the energy is supplied. Jones (8) confirmed this finding using a variety of match-head compositions. Bryan and Noonan ( 7 ) determined the minimum energies delivered in a 3-msec. interval that \Yere just sufficient to ignite a unit area of several explosives and found values which ranged from greater than 0.4 cal. per sq. cm. for T N T , to less than 0.1 cal. per sq. cm. for lead styphnate. T h e theory of ignition of solid combustibles has received detailed treatment (4,6, 7, 75-77), although the results. which usually require numerical integration techniques, are not generally discussed directly in terms of the ignition energy. In this paper the important features of the surface ignition energy and delay time are developed in a simplified manner from the heat conduction and decomposition kinetic processes involved.
Ignition Energy and Delay Time
When a step function heat source is brought into one-dimensional contact Xvith a unit area of a n explosive, heat flobvs into the explosive, raising its temperature. Although a finite time is required to establish an essentially constant temperature, T,qJat the interface between the heat source and the explosive, this time is often small compared to the ignition delay time of the explosive and will be ignored here. T h e neglect of this heating time is based on the fact that cxplosives are generally poor heat conductors) \vhich ensures a rapid 286
l&EC PROCESS DESIGN A N D DEVELOPMENT
The ignition energy dependence i s a function of the
establishment of a constant T , in the ideal situation. and that there is a large temperature dependence of the thermal decomposition rate of explosives, xvhich provides that the maximum temperature attained in the heating process will often very nearly determine the ignition time of the explosive. Calculations by Clarke (3)suggest the constant T,yapproximation to be reasonably good, if the ignition delay time is greater than several microseconds. T h e magnitude of T , for a finite heat source depends on several factors, including the temperature and energy content of the heat source, the intimacy of contact between the source and the explosive, and certain properties of the explosive. T h e one-dimensional transport of heat into the explosive (assuming no phase changes) is given by the heat conduction equation : pC
bT
- =
at
K
+ PQ ddtS
bZT -
3x2
-
T is the absolute temperature at position x and time t ; p . C, K , and Q are the density, heat capacity, thermal conductivity, and positive heat of reaction of the explosive; and d.\- 0 38 0 33 0 33 0 25
Computed ~~
E,b 0 4' 0 34
T". "K 940 738
0 28'
666
0 20'
55-
E , and 7 . ~~
E ~ , J. h
E, 0 41'
0 2-d 0 32e 0 23'
I , 859 654 694 506
T h e ignition energies, calculated by Equations 4 and 6 of several secondary explosives which were studied experimentally ~~
VOL. 4
NO. 3
JULY
1965
287
Table II.
Experimental and Theoretical Ignition Energies Using Equation 9
Exptl.
Ez.Q
Exjlosice
TNT Tetryl RDX PETN
Cal./ Sq. C m . >0.38 0 33 0 33 0 25
Computed E, and T ,
T", OK. 934 689
Et 0.46 0 30 0 26'
0 18h
E ~ , 4.9, T*
E, 0.36" 0 25d 0 30e 0 210
631 531
79 5
617 659 567
a From (I). Kinetics f r o m (5). ( 72). d ( 70). e (13). Autocatalytic, extrapolated to zero time. 0 ( 7 I ) . Autocatalytic.
Experimental data are not available to check the validity of Equation 10 quantitatively. Severtheless, the equation does predict that an increase in ambient temperature \vi11 decrease the ignition time. as is observed experimentally. T h e quantitative approach employed in this paper is that ignition is a pure thermal reaction. Ho\vever, it is knoivn that certain subsidiary factors such as chemical effects from certain gases, such as oxygen? or catalytic effects from the heating source may sometimes influence the ignition process. T h e inclusion of these factors is beyond the scope of this paper, but their general influence should decrease as the temperature of the source and its accompanying heat flux is increased. Acknowledgment
chemical reaction is pQL Z exp (-EIRTs), ivhere Q is the heat of chemical reaction and L is the length of the unit area column of explosive undergoing ignition. L may be approximated as the thermal thickness, kd:?J: and the thermal velocity, u, as L / t i . Thus, L = (kdt,)'I2. T h e heat flux through the surface of the explosive is obtained as before. Hence the ignition criterion is
K(T, - T o ) ,(Tkdt)"'
=
pQL Z exp(-EiRT,)
(9)
T h e ignition energies and surface temperatures calculated for the explosives of Table I using Equation 9 are given in ?'able 11. Q was taken as 500 cal. per gram in the calculations (76). T h e computed values in Table I1 are in reasonable agreement with those in Table I. T h e uncertainties in the values used for the physical constants, and in the ignition temperatures obtained by extrapolating Figure 7 of ( 7 0 ' ) , are too large to conclude unambiguously which table contains the more valid temperatures. I t is kno\vn experimentally that the initial ambient temperature of a n explosive may under suitable conditions influence its ignition delay time. Equation 6 does not provide for this effect. However, Equation 9 does include the effect of initial temperature. Thus
?he Lvriter extends his appreciation to 0. R . Irkvin and P. K. Salzman of Aerojet-General Corp., where this \sork \vas initiated, for their stimulating discussions on this subject. literature Cited
(1) Bryan, G. J.. Noonan. E. C., Proc. Roy. Soc. 246, 167 (1358). (2) Carslaw, H. S.. Jaeger, J. C.. "Conduction of Heat in Solids," chap. 11. Oxford University Press, London, 1959. (3) Clarke. J . F.. J . E'luidMech. 13, 47 (1362). (4) Cook, G. B.: Proc. Roj. Soc. A246, 154 (1958). (5) Cook, M. A , . .%begg. M. T.. Ind. Eng. Chem. 48, 1090 (1956). (6) Frank-Kamenetski. D. A , , "Diffusion and Heat Exchange in Chemical Kinetics, chap. VI. Princeton University Press: Princeton, 1955. (7) Hicks. B. L., J . Chem. Phys. 2 2 , 414.(1954). (8) Jones, E., Proc. Roy. Sot. A198, 523 (1949). (9) Morgan, J . D.. Phil. M a g . 49, 323 (1925). (10) Rideal. E. K., Robertson. A. J. B., Proc. Roy. Sot. A195, 135 (1948). (11) Kobertson. A. J . B., J . Soc. Chem. Ind. 6 7 , 221 (1948). (12) Robertson, A. J . B.. Trans. Faraday Soc. 44, 977 (1948). (13) Ihid.. 45, 8 5 (1949). (14) Stout. H. P.. Jones. E.. "Third Syniposium (International) on Combustion." p. 329. LVilliatns and LVilkins Press. Baltimore. 1949. (15) Yang, C. H., Combust. Flame 6, 215 (1962). (16) Zinn. J., Mader, C. L.: J . Appl. Phyr. 31, 323 (1960). (17) Zinn, J., Rogers. K. N.; J . Phys. Chem. 6 6 , 2646 (1962).
RECEIVED for review November 25. 1964 ACCEPTED February 16. 1965 Division of Fuel Chemistry, Symposium on Explosives. 145th Meeting, ACS,,New York. N. Y., September 1963.
AEROSOL FILTERS Pressure Drop across Single-Component Glass Fz'ber Fz'lters R
M
W E R N E R A N D
L. A. C L A R E N B U R G
Chemical Laboratory, .Vattonal Defence Research Organization T.VO. Rijsu ilk ( Z . H ). T h e .\-etherlands EVERAL
equations have been proposed to describe the
S pressure drop across fibrous filters for viscous
flohv.
From dimensional analysis of Darcy's law of flow through porous media. Davies (2) found an empirical expression relating pressure drop to filter porosity. T h e dimensionless group - W E 2 ~~
~
OLV is defined as the permeability coefficient! I;: in which p represents the pressure drop across the filter, '7 the viscosity of air, L the filter thickness. V the superficial velocity, and dE some 288
I&EC PROCESS DESIGN AND DEVELOPMENT
effective diameter. T h e effective fiber diameter is that fiber diameter \vith which the filter properties can be described as if the filter were composed of fibers of this diameter only. Davies' expression is given by
K = 64
x
(1 -
,)I5
[l
+ 56 (1 -
E ) ~ ]
(1 1
valid for fiber diameters ranging from 1 . 6 to 80 microns and filter porosities ranging from 0.700 to 0 . 9 9 4 . Davies stated that for high values of the filter porosity. e . the effective fiber diameter. d E . should be taken as the arithmetic mean fiber diameter. dB%.,of the fibers in the filter.
