Rapid Heating-to-Ignition of High Explosives. II. Heating by Gas Compression A. D. Randolph* and K. 0. Simpson Department of Chemical Engineering, University of Arizona, Tucson, Arizona 8572 1
The sensitivity of condensed-phase high explosives to accidental low-shock initiation by a gas compression mechanism was numerically studied by computer simulation with a rigorous thermalheaction kinetics model of the system. The validity of the model was verified experimentally.
Introduction The safe handling and utilization of high explosives requires a knowledge of their sensitivity to initiation when subjected to various external mechanical stimuli. An example of accidental ignition is when an explosive-filled shell explodes prematurely when accelerated during firing. Encased H E charges have also initiated when subjected to relatively small impacts, e.g., when dropped a few feet. Presumably, initiation in both of these examples was due to compression heating of gas in voids or cracks due to defective explosive loading. Although initiation is thought to be thermal in origin, several types of mechanical stimuli can result in localized concentrations of thermal energy sufficient to result in initiation of the explosive. Some postulated mechanisms for such mechanical to thermal degradation of energy are friction, strain heating, viscous dissipation, weak shock interactions, and gas compression. Sensitivity tests for explosives apply an external stimulus to an explosive sample, e.g., gap-attenuated shock, falling weight, skidding impact, or thermal shock, which thermally degrade through one or more of the above mechanisms to bring about initiation. Unfortunately, the actual mechanisms leading to initiation are poorly understood and different tests rank explosives in different orders of sensitivity. A more rational application of sensitivity test data to the practical problems of explosives safety might result if the mechanisms leading to initiation were better understood. Conceivably, different sensitivity tests could be developed which would better display a given initiation mechanism. P a r t I of this study analyzes the rapid heating-to-ignition by a friction mechanism for explosives subjected to oblique impact. The friction heating model which was developed, satisfactorily explained initiation data obtained with the large-scale skid sensitivity test. The present study analyzes the rapid heating-to-ignition of explosives by compression heating of gas in a confined void adjacent to the explosive surface. The thermal model developed in the present study was then used to analyze T N T surface ignition data obtained in a gap compression sensitivity test. In both cases, friction and compression heating, the thermal models which were developed, satisfactorily explained the initiation behavior of secondary H E solely as a runaway thermal pyrolysis reaction. Neither model attempts to explain the complex phenomena leading from ignition to violent explosion and/or detonation.
The Model The purpose of this study is to develop a realistic model of the ignition of condensed-phase HE when subjected to rapid heating by an adjoining compressed gas phase, e.g., in
an accidental drop of an H E device or during gun bore acceleration of a faulty shell. T h e goal in model development was to make a reasonable engineering model of the thermal events leading from gas compression to surface ignition which could be studied numerically to evaluate the response of H E ignition to various model parameters to better define conditions for safe handling of encased H E systems. The idealized physical gap compression ignition model shown in Figure 1 consists of three planar one-dimensional spaces, case/gap/HE or HE/gap/HE. In the model the gap is assumed to close until the plastic flow strength of the explosive is reached, a t which time gap closure stops. The confined gas is heated during compression and heat transferred by conduction to the walls. The model can be visualized as a piston closing with some velocity V ( t )until the plastic flow strength of the H E is exceeded; thereafter the piston is stationary relative to the deforming gap. No further heat is generated but the gap continues to transfer heat to the confining walls by conduction. Ignition is defined numerically when energy generated by H E decomposition in the first interior mesh point exceeds the rate of thermal transport through the mesh point boundary; the temperature then rises exponentially and the H E thermally “explodes”. The air gap is considered as a distributed system and continuity of temperature and thermal flux across the gap/ H E and gap/case boundaries is assumed. A constant mass (Langrangian) coordinate system with no flow between mesh points is assumed for the air gap. This assumption is based on dynamic pressure equilibrium in the gap, i.e., compression pressure waves traveling much faster than the velocity of gap closure. Figure 2 illustrates the physical assumptions of the model together with the equations that were used in the simulation. In the model the left phase, region 1, is an inert solid which is either a metal or high explosive. The equation for region l is the simple transient conduction heat equation. Region 2 is a gas-filled gap of initial width 6. T h e gas is heated by compression as the gap closes as represented by the dP/dt term. Region 3 is the reactive high explosive modeled by the addition of a reaction energy term in the transport equation. Thermal decomposition is assumed to occur according to zero-order Arrhenius kinetics (Zinn and Rogers, 1962). Thermal ignition is defined as the point where the temperature becomes high enough to initiate a runaway thermal reaction. All thermodynamic properties are assumed to be temperature dependent. A relatively new implicit differencing scheme (Keener, 1973; Keller, 1971) was adopted to solve the equations shown in Figure 2. The most striking feature of this scheme Ind. Eng. Chem., Fundam.. Vol. 15, No. 1, 1976
7
1.0-
HE COMPOSITE
CASE
WITH HIGH MECHANICAL STRENGTH
--
0.952
0.9-
0.8-
a)
C a s e / H E Separation Crack
0.7-
0.6-
4E
HE
0.4
-
o,zy/ b)
Internal Failure
In
PROFILES
HE
Figure 1. Idealized physical model of gap initiation mechanism.
1
0.I
V
0.0 0.0
0.2
0.4
d
0.6
0.8
i
1.0
t/tC
Figure 3. Dimensionless pressure rise curves.
CONTINUITY OF FLUX AND TEMPERATURE REGION I CASEIHE
-
will occur with the specified conditions using a Golden Section search routine (Wilde and Beightler, 1967). An internal (gas phase) pressure rise profile of the form
P ( t ) = (Pc - Po)(t/tc)”+ Po REGION I I C A S E I H E )
REGION 2 ( A I R GAP)
REGION 3 (HE)
Figure 2. NEWGAP ignition model. is the solution grid which is in the form of a box. This solution grid shape has several advantages over conventional difference methods which uniquely suit it for this problem. Briefly, the advantages of the “box” scheme finite difference method are t h a t it is implicit and stable for all values of the time and spatial step sizes. Arbitrary spatial step sizes are easily implemented and the scheme has secondorder accuracy a t all points. Nonlinear terms are treated by a first-order Taylor’s expansion and the solution iterated until convergence. Mesh points must fall on any discontinuities. Equation coefficients are evaluated a t the center of each solution grid. Thus the method is ideally suited for handling composite systems with large gradients near the phase boundaries. The “box” numerical technique applied to this thermal transport problem is given elsewhere in detail (Randolph and Simpson, 1974; Simpson, 1974). Numerical solution of the model equations was implemented on the University of Arizona’s CDC-6400 digital computer, as Program NEWGAP. Given a specific set of thermodynamic and physical properties the program solves the three transient heat transfer equations associated with an inert solid/compressed gasheactive explosive system in which heat is generated by compression of the gas-filled gap between the two solids. The program solves for both the temperature and flux profiles and provides automatic outputs a t plastic flow, ignition, and cooling. An energy balance is calculated and printed for each profile. The program determines the minimum gap size a t which ignition 8
Ind. Eng. Chem., Fundam., Vol. 15,No. 1, 1976
(1)
is assumed in the computations. P, is chosen as 1.05Pm,, for computational reasons. PO is the initial pressure, and P,,, and t , are the maximum pressure and time parameters characteristic of the system under study. This model encompasses the pressure response of physical events ranging from accidental drop to gun bore acceleration. Other profiles can be easily accommodated in the program, however. Figure 3 illustrates the pressure rise-time profiles t h a t can be modeled by eq 1. T h e generality of this equation is such that nearly any physical initiation event could be reasonably modeled with the given parameters. I t is emphasized t h a t the parameterization of the pressure profile, e.g., in eq 1, allows the subsequent thermal events to be decoupled in the computation from the original mechanical event.
Discussion of Results The numerical simulation which was developed in this study is a useful tool for analyzing data from sensitivity studies as well as realistically predicting conditions for gap initiation in weapons systems. These calculations indicate that non-shock gap compression heating is a viable mechanism for ignition of encased high explosives. A number of thermodynamic and physical properties of explosive (case)/gas/explosive systems were studied to determine their effect on explosive sensitivity. The variables studied included the effects on sensitivity of explosive thermal conductivity and plastic flow strength, a high conductivity inert wall (steel case), initial explosive temperature, compressive events typical of drop or gun bore acceleration, and explosive reactive properties typical of RDX or HMX (see Table I). Figure 4 plots typical ignitionho ignition separatrix curves assuming properties typical of RDX explosive with an air gap. As would be expected, a smaller air gap is pre-
Table I. Properties of Solids Used in the Simulation _________
Solid -__
h, W/m K
0.424 0.106 51.90
HMX RDX Carbon steel
C,
_
_
~
AH,, k J / k g K p , Mg/m3 MJ/g-mol A, g-molisec mz
______
_____
1.05
1.91
1.05
1.81 7.85
0.448
2.77 2.10
___
________
E*, MJ/ g-mol
9.81 x 1 0 2 2 5.66 x l o 2 ' - - _ _ _ - _- -
Reference
0.218 0.197
Randolph et al., 1972 Randolph et al., 1972 Perry, 1963
_____
1.20-
1
1.00-
o'60 I0.50-
50.80-
0
-n= 5.0
L n m w 0.40
-
60.60ln W
z Y
z
0
g0.40-
1
I-
X
a 0.30-
c0.30-
U
U
I025-
IC = I.5mSEC pO=l B A R PC=3.15 K B A R PMAX = 3.0 K B A R
0.203
I
zI0.20- -
rI 0.15-
Pc=315 K B A R : , . ,P ,
I..^"
z
3.0 K B A R
Po=l B A R To: 2 9 4 . 4 K
OS
07
09
15
20
30
40
60
80 IO0
PRESSURE EQUATION E X P O N E N T , N
Figure 6. Effect of the pressure-profile equation exponent on the minimum gap thickness for ignition.
0'601
0.50
I 040u W-
5 030a 4
0
025-
I
zz
z
H MX /AIR / H M X
0.4
03
, 05
5 oeo-
s
Pc=3.15 K B A R PMA,=3.0 KBAR pO= I B A R T0'2944 K
PHAX= 3.0 K B A R P ~ p ~ = 5 K. B0A R Po= I BAR To= 294.4 K
I
06
07
08 09 IO
CHARACTERISTIC T I M E
15
2 0
MILLISECONDS
Figure 5 . The effect of the pressure-time profile on the thermal ignition of HMX. dicted to ignite the H E as the compression time (to the same maximum pressure) decreases. T h e parameter n indicates the shape of the pressure rise profile as shown in Figure 3. The effect of the shape of the pressure rise-time profile as presented in Figures 4-6 was significant. An unexpected maximum in the minimum gap thickness plots as a function of the pressure rise-time equation exponent, n , was observed. This apparent effect of pressure rise might be one of the contributing factors in the wide disparity in results obtained in different test apparatus; a sensitivity test simulating a particular event should strive to match the pressure rise-time profile of the real process. For instance, use of a gap compression apparatus such as that utilized by Bryan and Noonan (1958), which develops a pressure profile characteristic of free fall, might be misleading in other situations. In a free fall the acceleration is constant and most of the pressure rise occurs in the last 20% of the compression time interval (see the n = 10.0 profile in Figure 3). Contrast this to the profile developed during gun bore acceleration. Here the acceleration is initially high and decreases with time. Pressure rise-time profiles typical of gun bore acceleration can be modeled with pressure rise-time
0.12
0.5
0.6 0.7 0.8 0.9 1.0 1.5 CHARACTERISTIC T I M E , MILLISECONDS
20
Figure 7. Effect of ultimate flow strength of the high explosive on minimum gap thickness for ignition. equation exponent values, n , less than or equal to unity (DeVost, 1973). The parameter t , is a measure of the time scale of each type of event. For example, the characteristic time of closure for accidental drop is directly related to the drop height and the gap width. Figures 4 and 5 show that the critical gap sizes for a characteristic time of 0.5 msec are 2.26 mm for RDX and 3.73 mm for H M X ( n = JO.0). If a complete gap closure is assumed (drop velocity is given as V= and the critical time is the initial gap width divided by the velocity) then the minimum drop heights above which an encased weapon containing the explosive is unsafe are 1.04 m for RDX and 2.96 m for HMX. Alternately, an RDX filled system containing a separation crack of 3.73 mm would be unsafe if dropped above a height of 33.0 cm as compared to 2.96 m for HMX. The effect of plastic flow strength on explosive sensitivity is shown in Figure 7 . This effect is seen to vary with the pressure rise-time exponent, n , being more critical for gun bore acceleration (n less than or equal to 1.0) than for accidental drop ( n > 10.0). Sensitivity increases with plastic flow strength as total gas confinement pressure in the
a
Ind. Eng. Chern., Fundam., Vol. 15, No. 1, 1976
9
0.60-
-R D X / A I R / R D X
---
CARBON STEEL/AIR/RDX
0.50-
I U
.0.40N
a 0 030-
I
= I I 0.20
z
I /
5
=0151
? E
I
,
,
0.5
0.6
,
PHAX=3.0 K B A R Po = 1 BAR n = 10.0
, , ,
Pc:3.15 K B A R PMAx= 3.0 K B A R Po'l BAR
0.20-
I 1.5
0.7 0.8 0.9 1.0
025-
018-
,
,
,
,
1
,
To=2944 K i
CHARACTERISTIC TIME, MILLISECONDS
Figure 8. Effect of initial temperature on the minimum gap size for ignition.
OdO1 0.500
0.40 0,501
m- 0.40-
.
(0
z 0 0.30P
4 0 0.25-
I 5
Pc=3 I 5 K B A R P M A x = 3 0K B A R Po= I B A R T0'2944 K n =10.0
0.20-
I
I 05
0'6
07
0'6 0'9 10
CHARACTERISTIC
15
20
T I M E MILLISECONDS
PMeX = 3 0 K B A R To = 2 9 4 4 K
Figure 9. Effect of thermal conductivity on the minimum gap size for ignition.
model varies directly with the physical strength of the HE composite. Calculations on the effect of initial system temperature indicate that any significant temperature rise increases the sensitivity of the explosive. While the 94.4 K temperature rise chosen as a random example may be larger than those encountered in typical weapons systems, interpolation of the results shown in Figure 8 show t h a t even a 20-30' temperature rise produces a significant rise in sensitivity. If a defective projectile containing a crack were left in a hot gun barrel for some time before firing, the temperature rise could increase the sensitivity to the point where the gap was thick enough to cause ignition. However, a n increase in temperature would also decrease the plastic flow strength. Therefore, the effect of temperature may not be as pronounced as these results indicate. Apparently the most important variable affecting explosive sensitivity is the thermal conductivity of the explosive. Figure 9 illustrates the profound effect of this variable. Anisotropic transport properties, e.g., low conductivity crystal axes, would increase the explosive sensitivity markedly. If the crystals assumed a random orientation in the composite HE, ignition would be predicted t o occur at "hot spots" formed by crystals with a low conductivity axis perpendicular to the plane of the gap. A lower thermal conductivity results in temperatures a t the gas-explosive boundary that rise faster, given the same pressure rise profiles. Thus a smaller gap would suffice to produce the necessary temperature rise on the explosive surface. Thermal transport measurements with large single crystals are needed t o determine if such anisotropy of conductivity exists and its possible importance in explosive sensitivity. T h e higher thermal conductivity of H M X is the predom10
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
inant reason for its lower calculated sensitivity to compression ignition. T h e higher heat loss to the explosive prevents the temperature rise necessary for ignition unless a larger gap is used to provide more capacity. Note the shift in position of the n = 0.5 lines for RDX and HMX relative to the n = 10.0 lines in Figures 4 and 5. For n = 0.5 the major part of the energy generation occurs a t the beginning of the compression. Thus, heat losses from the gap into the explosive during the period of highest energy generation would tend to desensitize the explosive more for a slow initial compression rise time profile than for a later rapid compression. T h e typically low thermal conductivity of condensed phase H E is the most important single factor causing its sensitivity to gap compression. Significant desensitization of composite H E to this ignition mechanism might be achieved by intimate dispersion of a high conductivity, but energetic, material in the composite matrix. However, dispersion would have to be on the same size scale as the temperature gradients formed during compression, say of the order of a few microns. Figure 10 compares the sensitivity between case/gap/HE and HE/gap/HE configurations. T h e higher thermal conductivity of the metal case desensitizes the system by acting as a heat sink to decrease the energy available for raising the temperature of the explosive. This effect is relatively independent of the pressure rise-time profile, but is more pronounced a t longer compression times t,. This heat sink potential is ultimately limited by the ability of the gas to conduct energy to the metal. Of greater importance is the thermal conductivity of the explosive since its ability to transfer heat determines the
Table 11. Thermal Properties of Various Gases Used in the Simulation Casu
Air Argon Helium Nitrogen Oxygen (1
C,, kJjkg K
P,kg/m3
Y
a
Reference
25.62 11.48
1.00 0.523
1.20 1.65
1.40 1.668
0.823 0.762
142.12 25.44 26.14
5.19 1.04 0.921
0.165 1.16 1.32
1.66 1.40 1.395
6.655
Kreith (1964) Perry et al. (1963) Rohseno w (1973) Kreith (1964) Kreith (1964) Kreith (1964)
h, mW/m K
0.717
0.830
Properties evaluated at 294.4 K and 1.0 bar.
temperature rise a t the explosive--gas interface. For large gap sizes any wall effects will probably disappear. In studies which have reported a decrease in sensitivity attributable to a change in a boundary material the explosive has been in contact with the metal. The Naval Research Laboratory attributed the change in sensitivity in their impact tests to a higher heat loss from the compressed occluded gases in the explosive (Bowers e t al., 1973). Picatinny Arsenal found that the sensitivity of the explosive to void compression was decreased if the metal punch of their actuator was painted (Schimmel and Weintraub, 1970). No explanation was given for this phenomenon and insufficient information was given to test it in this model. The effect of the nature of the confined gas on sensitivity as shown in Figure 11 was studied in more detail since some research had been done in this area. Evans and Yuill (1953) used a n apparatus in which a gas was compressed over an explosive by action of a ball dropped on a piston. T h e piston was not locked into place a t the end of the compression but was allowed to rebound. Compression times were of the order of 5 msec with gaps of approximately 3-4 cm in length. T h e rebound rate was of the same order as the compression rate. The compression velocity for each test was constant and the compression ratio was progressively increased until ignition occurred. Two high explosives, nitroglycerine and P E T N , were tested in atmospheres of oxygen, air, nitrogen, and argon. They found t h a t oxygen was more efficient than air which was in turn more efficient than nitrogen for ignition. They concluded t h a t the chemical nature of the gas is important, indicating a gas/HE reaction may take place. Since nitroglycerine and P E T N are among the most sensitive of the high explosives this speculation is probably valid, especially for nitroglycerine which can vaporize easily thus allowing reaction in the hot gas. Their conclusions on the ignition efficiency of argon and nitrogen was t h a t “For a given compression ratio, argon is more efficient than nitrogen, because of its higher ratio of specific heats” (see Table 11).This conclusion is confirmed by the results of this study. For a given compression ratio (3000:l) argon is much more efficient for thermal ignition than air. Comparison of the results in Figure 11 shows t h a t a t a characteristic time of 1.5 msec, the minimum gap for argon is 2.56 mm and t h a t for air is 3.78 mm. A velocity profile representing dropping of the system ( n = 10.0) was used since it also approximates a constant velocity of compression. Air was assumed to be inert and thus can be compared to nitrogen in Evans and Yuill’s test. Helium was also chosen for this study because its thermal capacity (pC,) and heat capacity ratio (see Table 11) are approximately equal to those of argon. However, the thermal conductivity of helium is approximately 8.5 times that of argon. In order to compare the heat transfer processes occurring during compression, runs were made for the same pressure rise--time profile ( n = 10.0, t , = 1.5 msec) with each gas. An HE/gap/HE crack in RDX was assumed. T h e runs were made a t the critical gap size for each
gas; the compression ratio is thus the same for both gases. The critical gap sizes were 2.56 mm and 3.96 mm for argon and helium, respectively. A detailed comparison of the temperature profiles and thermal fluxes leading to ignition for the cases of helium and argon showed t h a t the thermal conductivity of the gas was the most important factor determining sensitivity. However, t,otal thermal capacity limited the effectiveness of ignition with helium, even with its large thermal conductivity. T h u s the high energy loss from the helium during the early stages of compression required t h a t a larger thermal capacity, i.e., larger gap size, was needed to provide enough energy to offset the losses later in the compression when thermal generation is much smaller. The results in Figure 11 show t h a t air falls between helium and argon in ignition efficiency, but closer to helium. The thermal capacity ( p C , ) of air is slightly larger than that of argon, the thermal conductivity slightly larger, and the heat capacity ratio is considerably smaller. Thus, the reason for the lower ignition efficiency of air vis-a-vis argon is the lower heat capacity ratio. T h e maximum temperature which can be developed in air is much smaller than t h a t of either argon or helium. This is offset somewhat by the larger thermal capacity. T h e result is a complex interrelationship of properties which cause the final positioning of air vis-a-vis argon and helium in the critical gap vs. t , curves.
Experimental Study of Gap Ignition Mechanism
A. Description of Experiments. Data from a gap sensitivity test conducted a t Los Alamos Scientific Laboratory were analyzed by the NEWGAP simulator in verification of the gap compression ignition mechanism. The sensitivity test used was the Aquarium technique originally developed to study shock initiation (Eyster et al., 1949). T h e pressuretime gradient transmitted to an explosive sample by this technique is drastically different from that experienced, for example, by an artillery projectile in actual launching. However, the technique is geometrically simple and is anienable to simulation using the one-dimensional planar thermal ignition simulator discussed previously. With the Aquarium technique, a sphere of P B X 9404, 25.4 mm in diameter, and several samples of test explosive a t an appropriate radius were submerged in water. T h e sphere of P B X 9404 was initiated near its center with mild detonating fuse (MDF). Depth of submersion was chosen so that the gas bubble vented shortly before it reached its maximum radius, which prevented oscillation of the bubble so that only a single shock wave was generated. Support, confinement, and base gaps for the samples of test explosive were provided as described below. The peak pressure of the shock wave transmitted to the water is a function of radius. Thus, the magnitude of the compression wave could easily be adjusted by positioning the H E samples a t different radii. T N T pressed to a density of 1.611 & 0.001 g/cm3 (97.4 wt % of theoretical density) Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
11
Table 111. Thermal Properties of Various Gases -Aquarium Tests
Gas
Air Methane Krypton
Heat capacity, Jie K 1.0
Conductivity Wlm K 0.026 0.034 0.0099
2.22 0.247
29 16
83.8
it experienced a higher peak pressure. The amount of material consumed appeared to he proportional t o t h e thickness of the air gap. None of the samples detonated, presumably because t h e arrangement did not confine the products adequately t o lead to a transition to explosion. Apparently, the sample with t h e 15-mm air gap almost exploded as evidenced by considerable breakup. T h e experiment demonstrated that gap compression is a viable mechanism for H E ignition when an encased system containing a separation gap is subjected to low-shock rapid compression of the gap. These experiments, and a repeated set giving similar results using a slightly different mechanical design of the Plexiglas air gap assembly, indicated that t h e T N T surface would ignite under these low-shock conditions with air gaps of 1 mm and greater. No tests with gaps less than 1 mm were attempted. Another series of experiments was made using methane ..were cnwen .,. ...~~ and krypton in the adjoining gap. These gases because of their wide variation in thermal properties. Table 111 compares the thermal properties of the se two gases with air at armtjspheric preswre and remperatuire. lanition d ‘ t h c TYI’ curinre with krvi)ton and merhanr “. gaps occurred in all samples, even with the minimum 1-mm gap. A shielding experiment was next performed in which a l i l of ~ l i i m i n n m w s r olnrrl tn TNT nrfsra 13-wm thick fc.. ”_ tho ”___ __.cI__.___ by a thin layer of Eastman 910 cement (idt h c sphrre oi I’HN 9.101, experienced a peak shock prrssure oi npproximately :iklwr+: wirh a moderilte air gap, t h r peak pressurr IW; reduced 1 3 alimit ? khar-. in The
vir.
Fixlire 12 IIIIIWC II typical aaseniblt uwd in the- Aquarium . I n undrrsized O-ring was plwrel near one end d i d 1 sample. A I’lexigla. cylinder IS slipped over eat h 0 ring and glued i t , that i t rxrended 3 prr-elected distance herwid tlii, TXT. 1’1exighs diiki (1.3 mm rhicki \wre iitied i n J i , t h r exrended ends o i t h e I’lexi ‘iir gal,. This impro\rd the -hock impedance niat,.h w that the snmple experienced n peak pressure ne’iir 4 kliarh. Figure 1:I is a phzBtqrapn U I rtnwered TYI‘ Stsic that nll samples which had air z i p + lpartially hiirned. ‘ h r one ianiple without a gap showed 1181 i,ridi,ner ( 1 1 hurn althouzh 1e.t-
12
nd. Fng Chem.. F.ndam, V o . 15. ho 1. 1976
_..”..-
~...~.~ “..=
k
h i s function was input as a user-supplied subroutine in
Program NEWGAP. Thermal conductivity of methane was obtained from
Table I V . Physical and Thermochemical Properties Used in Simulation of Aquarium -~ Tests Range of Property Symbol values
Thermal conductivity, W/m K Heat capacity, J / g K Density, kg/m7
~
~
Gas
TNT Thermal conductivity , W/m K Temperature coefficient of conductivity, K-l Heat capacity, J / g K Density, kg/m3 Activation energy, J / g mol Heat of reaction, J/g mol
Table ,V. Calculated Minimum Gap Thicknesses Kequired t o Ignite TNT in the Aquarium Test (nominal Thermal Properties for TNT)
0.13-0.26
_
_
p,,,,,, MPa
340 200 203
6'rl*, mm 1.58
tcxpr1,
psec
-1.5 x 103-0
Air Krypton Methane
AE+
1.37 1.65 X l o 3 1.72 X 10'
Table VI. Calculated Minimum Gap Thickness Required to Ignite TNT in the Aquarium Testa
AH
105
k, 0 3
c 3 P3
Gas Plexiglas h,
0.21
c,
1465 1149
PI
data of Owens and Thodos (1958) a t 1300 K and 200 MPa. From their correlation, methane thermal conductivity a t these conditions is
h = 7.8hc = 0.396 W/mK This constant value of conductivity was used for all methane calculations. The perfect gas relationship was used to calculate gas density and hence gap gradients. Significant errors in thermal gradient may have been calculated for methane by this technique as reduced methane temperatures were considerably lower than for air or krypton. C. Simulation Results. T h e Aquarium shots described in section A were simulated using Program NEWGAP. As mentioned previously, the rapid microsecond pressure risetime of these experiments is grossly different from the characteristic millisecond setback pressure rise-times measured with artillery shell or typical of an accidental drop. Nevertheless, these Aquarium shots were used to test the gap ignition mechanism as implemented in Program NEWGAP.
T h e probable pressure-time profile t h a t the gap experienced was calculated independently using the S I N hydrodynamic code. (Group T-4, Los Alamos Scientific Laboratory, Los Alamos, N.M.) These calculated pressure-time values for air were empirically expressed in the form of eq 1, namely
T h e following parameter fits were obtained: t , = 1.15 bsec; n = 11.4; P,,, = 340 MPa. Calculations revealed t h a t the parameters t , and n had essentially no effect on calculated ignitions, while the total compression P,,, had a marked effect. T h e values of t , and n calculated from air were then used for methane and krypton. T h e calculated values of peak pressure were 340, 200, and 203 M P a for air, krypton, and methane, respectively. Table V shows the calculated critical gap thickness to produce ignition of the T N T surface. Nominal room temperature thermal properties of T N T shown in Table IV were used in these calculations. These calculations indicate t h a t the calculated critical gap thicknesses for air and krypton correspond within experimental error with the smallest experimental gap thicknesses tested of 1 mm which caused ignition a t the T N T surface. However. the calculations for methane indicate a
pnlax,
MPa
1.39 >15
7.48
24.7 >177
6crir7
mm
Air 340 0.44 (0.28) Krypton 200 0.24 (0.16) 2.92 (1.2) 203 Methane ah,'= 0.13 W/rnK; a , = -0.5 X 1 0 K - ' .
1.73 1.73 14.4 (2.88)
much larger critical gap size than experimentally required to ignite the samples. Further, the calculated times to explosion for krypton and methane are longer than the calculated pressure duration in the Aquarium shots. A second set of calculations was made with reduced thermal conductivity of T N T ( h = 0.13 W/mK) and a negative conductivity coefficient of -0.5 X 10- K-I. Thermal conductivity can be a highly anisotropic property in high molecular weight organic crystals; the measured pressed or cast bulk thermal conductivity would then be an average taken over the random orientation of all single crystals in the composite. However, the thermal sensitivity of the H E would be determined by any crystals with a low conductivity axis exposed to the adjacent gap. These crystals would form "hot spots" for the beginning of ignition. T h e existence of such anisotropy would have to be demonstrated from experimental measurements of conductivity along different crystal axes in large single crystals of HE. Such anisotropy is suggested here as a possible mechanism of thermal concentration in these experiments. The conductivity temperature coefficient of -0.5 X l o - ' K-I is less than the -1.8 X lo-' K-' value obtained from a gross extrapolation of temperature dependence measured near ambient conditions; this value compares well with the (-0.5 to -2.0 X 10-.')K-' temperature coefficient range reported for various liquids (Reid and Sherwood, 1966b) and should be a conservative estimate of the temperature-dependent decrease in T N T conductivity. Table VI shows the calculated minimum gap thicknesses for the three gases using the lowered and temperature-dependent T N T thermal conductivity. The Arrhenius decomposition kinetics, determined from analysis of induction times of the order of seconds, represent an extrapolation of more than six orders of magnitude to the time of scale of the Aquarium tests. The critical gap heights and explosion times shown in parentheses in Table VI were calculated using a tenfold increase in the preexponential factor ( A = 1.12 X 10l8 g-mol/sec mj). Using this reaction rate constant together with the low estimate of T N T thermal conductivity, Table VI suggests that the bare-surface Aquarium shots can be explained with the one-dimensional gap model. The calculated order of initiation of the three gases is in the same order as their heat capacity ratios, with krypton (y = 1.67) being the most efficient initiator. Although methane did not develop high calculated gas temperatures (T,,, ca. 1700 K), its high thermal conductivity and heat capacity provided efficient heat transfer leading to ignition. If the gap ignition model is accepted as the explanation of these data, the lowered T N T Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
13
Table VII. Maximum TNT Boundary Temperature in Aluminum Foil Experiment0 Foil thickness, pm 13 7
4
TNT temp, K 5 36 65 6 794b
0 k,O = 0.13 W/mK; a 3 = -1.2 x at point of ignition.
tmax
psec___
192 129 102 K-I. b TNT surface
Table VIII. Critical Gap Thickness to Ignite TNT with 4-pm Aluminum Foil0 Pmaw
MPa
~
340 5.9 500 4.4 a h o = 0.13 W/mK; a 3 = -1.2 X
193 88 K-I.
conductivity and increased decomposition rate appear as reasonable parameter choices because they result in ignition times well within the duration of the pressure pulse. In one of the Aquarium experiments, a 13-pm aluminum foil was cemented against the HE surface. Compression with a 5-mm air gap produced ignition a t the center of the H E disk (with rupture of the aluminum foil) while a 1-mm gap produced no reaction. This experiment provided the only critical evaluation of the compression heating mechanism, in the sense that the “go”-“no-go” boundary was spanned with variation of air gap thickness. Program NEWGAP was modified to simulate the insertion of an inert phase against the H E surface, thus simulating the aluminum foil experiment. Calculations using a 5-mm gap compressed to 340 MPa showed that most of the compression energy was transferred to the high-conductivity aluminum strip, which then slowly transported heat to the H E surface until it decomposed or cooled. The entire time scale of the process was slowed by nearly two orders of magnitude to the order of fractional milliseconds. In effect, the rapid compression heating and transfer to the foil deposited a hot aluminum strip on the HE a t time zero, which then slowly heated the H E to ignition or eventual cooling. Expansion of the hot gases in the rarefaction wave was not considered in the calculations. Rarefaction cooling would have little effect on the overall heat transfer as the gases would probably remain as warm as the aluminum foil and the gradient for heat transfer would be destroyed by the expansion. The aluminum foil proved to be a large heat sink relative to the energy generated by gap compression. Table VI1 shows the maximum calculated H E surface temperature as a function of assumed foil thickness after the compression of a 5-mm air gap. These calculations indicated that the aluminum foil acted as a large heat sink relative to the total amount of compression energy. Ignition was limited by total available energy, not transport rates. Calculations assuming a 5-mm gap showed t h a t ignition temperatures could not be achieved with foils of 5-pm thickness or greater. A critical gap search was made with an assumed 4-pm aluminum foil and peak pressures of 340 and 500 MPa. Table VI11 lists these results. In the calculations in the table, the higher value of the preexponential factor was used. These calculations indicate that the 13-pm aluminum foil should have acted as an efficient heat shield to protect the T N T surface from ignition. The fact that ignition was 14
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976
observed experimentally with the 5-mm air gap indicates some concentration of energy. A possible explanation is that the Plexiglas disk closed on the air gap in a way to form an air pocket over the center of the aluminum/HE surface, thus creating additional thermal capacity over that location. As mentioned previously, total thermal energy, not transport rate, appeared to determine ignition with the aluminum foil tests.
Summary and Conclusions The one-dimensional, three-media, transient reactive heat transfer model, as implemented in Program NEWGAP, satisfactorily explains the bare-surface T N T ignition observed experimentally in the Aquarium tests if reasonable assumptions are made for the thermal transport properties and reaction parameters of the system. Surface ignition which was observed with a thin aluminum foil on the T N T surface and a 5-mm air gap was predicted with the model only when the foil thickness was assumed to be reduced by two-thirds. Ignition with the 13-pm aluminum foil can only be explained with this thermal transport mechanism by assuming a concentration of thermal capacity in the gas behind the point of ignition. Nevertheless, the foil-covered shots could best be understood by a thermal transport model and should prove useful in further experiments to definitively characterize ignition mechanisms. The NEWGAP ignition model was used to simulate ignition in encased H E systems with thermal properties representative of RDX or HMX explosives. The model was used to study the predicted effect of gas properties, H E thermal conductivity, pressure rise-time profile, and initial temperature on ignition by the gas compression mechanism. Ignitions were predicted for the kinds of gap separations characteristic of faulty H E systems and the pressure rise-time profiles experienced in accidental drops and/or gun bore acceleration. The Aquarium tests, their theoretical analysis, and calculations with the NEWGAP ignition model extended to encased H E systems indicate that compression ignition of H E adjacent to an air gap is a viable mechanism that could lead to accidental initiation of the H E charge. Acknowledgments This study was funded by AFOSR Contract F44620-70C-0080. The Aquarium experiments were performed a t the Los Alamos Scientific Laboratory under funding provided by the Defense Advanced Research Projects Agency, Order No. 2502. The authors are indebted to Mr. B. G. Craig of Group M-3, Los Alamos Scientific Laboratory, who conducted the Aquarium sensitivity tests. Nomenclature A = Arrhenius frequency factor a = coefficient in lz = ko (1.0 a ( T - To))relationship C, = constant pressure heat capacity E*,EI = activation energy in Arrhenius expression H3 = heat of reaction of H E h = thermal conductivity P = pressure R = gasconstant R B = decomposition rate of H E T = temperature t = time = velocity V = specific volume x = spacecoordinate 6o,6 = gap width 8 = dimensionless temperature 7 = dimensionless length
+
p T
= density = dimensionless time
Subscripts 1-3 = phase number in simulation Superscript 0 = original conditions Literature Cited Bowers, R. C.. Romans, J. B., Zisman, W. A,, lnd. Eng. Chem., Prod. Res. Dev., 12, 2 (1973). Bryan, G. J.. Noonan, E. C., Proc. Roy. SOC. London. Ser. A , , 246, 167 (1958). DeVost. A,, "Premature Simulator for Projectile Explosives" (Progress Report No. 3), NOL Tech. Report 73-52, May 1973. Evans, J. I., Yuill, A. M., Proc. Roy. SOC.London, Ser. A, 246, 176 (1968). Evster. E. H.. Smith. L. C.. Walton. S. R.. "The Sensitivitv, of Hioh -,- Exolosives to Pure Shock". NOL Memorandum 10, 366 (1949). Keener, J. P., Department of Mathematics, University of Arizona, Tucson, Ariz., private communication, 1973. Keller, H. B., "Numerical Solutions of Partial Differential Equations", B. Hubbard, Ed., Proceedings of 2nd Symposium on the Numerical Solutions of Partial Differential Equations, SYNSPADE, 1970, pp 327-431, Academic Press, New York, N.Y., 1971. Kreith, F., "Principles of Heat Transfer", International Textbook Co., Scranton, Pa., 1964. ~
Owens, J. E., Thodos, G.. "Reduced Thermal Conductivity Chart for Methane" Proceedings of the Joint Conference on Thermodynamic and Transport Properties of Fluids, Inst. Mech. Engr., Westminster, London, 1958. Perry, R. H., Chilton, C. H., Kirkpatrick, Ed., 4th ed, "Chemical Engineers' Handbook", McGraw-Hill, New York, N.Y., 1963. Randolph, A. D., Hatler, L. E., Popolato, A,, "Effect of Target Thermal Conductivity and Roughness on the Initiation of PPX 9404 by Oblique Impact", Los Alamos Scientific Laboratory Report LA-4843, Mar 1972. Randolph, A. D., Simpson, K. O., "Study of Accidental Ignition of Encased High Explosive Charges by Gas Compression Mechanisms" AFOSR-TR74-1239, Gov. Accession NO. AD783296 (1974). Reid, R. C., Sherwood, T. K., "The Properties of Gases and Liquids", 2nd ed, p 477, McGraw-Hill, New York, N.Y., 1966a. Reid, R. C., Sherwood, T. K., "The Properties of Gases and Liquids", 2nd ed. p 505, McGraw-Hill, New York. N.Y., 1966b. Rohsenow, W. M., Hartnett, J. P., Ed., "Handbook of Heat Transfer", McGraw-Hill, New York, N.Y., 1973. Schimmel, R. T., Weintraub, G., "Effect of Base Separation and Loading Density in the Setback Sensitivity of Composition A3," Picatinny Arsenal Tech. Report 4147, Nov 1970. Simpson, K . O., Ph.D. Dissertation, University of Arizona, Tucson, Ariz., 1974. Wilde, D.J., Beightler, C. S., "Foundations of Optimization", p 242, PrentlceHall, Inc.. Englewood Cliffs, N.J., 1967. Zinn, J.. Rogers, R. N.. J. Phys. Chem., 66, 2646 (1962).
Received for recielc January 9, 1975 Accepted July 31, 1975
A General Method for the Calculation of an Ideal Cascade with Asymmetric Separation Units D. Wolf, J. L. Borowitz,' A. Gabor, and Y. Shraga Isotope Department, The Weizmann lnstitute of Science, Rehovot, Israel
The theory of ideal cascades for the separation of isotopes has been reexamined for the case where the heads and tails separation factors of the elementary separating unit are not equal. Starting from basic isotope separation equations, a method for the calculation of flows and concentrations is described, which is applicable to an ideal cascade based on an elementary unit with any ratio between heads and tails separation factors. This method also serves to solve for any ideal cascade with any number of sidestreams.
Introduction T h e theory of isotope separation for ideal, square, and squared-off cascades has been thoroughly developed in the past three decades (Cohen, 1951; Dostrovsky, 1952; Lehrer, 1961). While most of the published work on ideal cascades refers to symmetrical separating units, few publications can .be found on the design of cascades with asymmetrical units. Pratt (1967) in his hook gives the solution for the concentration profiles in an asymmetrical cascade. Apelblat and Wolf (1972) suggest a solution for concentrations and flow profiles in a particular type of asymmetrical cascade, and Takashima (1965) presents a complete solution for flows and concentrations for a n asymmetrical cascade with the only limitation being t h a t i t requires particular ratios hetween the heads and tails separation factors. In this paper we present a general method for calculating all the flows and concentrations in an ideal cascade based on an elementary separating unit of any asymmetry. All stages, however, are assumed to have the same separation factors. T h e method is applicable also for the case of any ideal cascade with various feed streams a t different concentrations and any number of side products at different concentrations. Industrial use of an isotope separation cascade hased on asymmetrical separation units has been proposed
by Geppert e t al. (1975) for the separation of uranium isotopes using the separation nozzle method.
The Elementary Separating Unit A diagrammatic representation of a n elementary separating unit is given in Figure l. Feed a t a rate F, and a concentration X F , is fed to the unit. This feed is separated into a n enriched stream V,, a t a concentration Y,,and a depleted stream L , a t a concentration X,. T h e cut 8, is defined as V,/F,. T h e overall separation factor for the unit is CY and is given by cy=-
Yr X, 1-YrIl-Xr
T h e heads separation factor,
(1)
is given by
and the tails separation factor, P a by (3) clearly
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1 9 7 6
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