Equation of state of detonation product gases

EQUATION OF STATE OF DETONATION PRODUCT GASES E. L. LEE AND H. C. IIORNIG

Lawrence Radiation Laboratory, University of California, Livermore, California We have measured the Chapman-Jouguet adiabat for PETN at p0 = 1.77 g/cc and have combined our results with published data on the dependence of detonation velocity on density to determine the value of the Grfineisen ratio O as a function of volume for detonation product gases. We find G(V) exhibiting a broad maximum, value 0.6, at a gas density of 2.0 and decreasing to values between 0.2 and 0.3 at gas densities below 1.0. We have generated an equation of state for PETN using G(V) and our measured adiabat and have calculated CJ parameters including temperature and adiabatic expansions at various loading densities. Pressures are somewhat higher than those measured by Dremin, I and somewhat lower than measured by the Los Alamos group. 2 measured dependence of detonation velocity on loading density we will show how a general equation of state may be obtained. We chose to study P E T N in hopes of distinguishing between effects due to the equation of state of the fluid and effects due to chemical reactions. Since this explosive is relatively oxygen-rich,

Introduction Of the many empirical equations of state which have been described in the literature a the most widely used for high-explosive product gases has been the so-called polytropie or gamma law equation which takes the form,

pVr= K

(1) CsHsN40,2--+4 H~O+3 C02+2 C 0 + 2 N2

for the adiabat. Fickett and Wood 4 found that with P equal to approximately 2.7, Eq. (1) described the measurements made by Deal 5 of the dependence of particle velocity on pressure for the adiabatic expansion of the detonation products of Comp B, Grade A. They also showed that if Grtineisen's ratio G for which they used the symbol fl-~ was assumed constant, they could calculate the measured dependence of detonation velocity on loading density. In a recent paper 6 we have shown that more accurate measurement of the adiabatic expansion cannot be described by as simple an expression as the polytropic equation. The purpose of this paper is to investigate the effect of the improved adiabat information on the equation of state by examining its effect on the Griineisen ratio. We will assume that the Chapman-,louguet hypothesis is valid and that dG/dT is negligible over the region of interest. By applying our information to the prediction of adiabat expansion experiments for a variety of loading densities, we can eventually test our assumptions. We will show here that Grfmeisen's ratio cannot be constant, but must, at least, be a function of the volume. By making use of the previously measured adiabat for PETN, O0= 1.77, and the

shifting equilibrium involving formation of such substances as carbon and methane should be of minor importance. P E T N is amenable to lowdensity pressing and is reasonably safe to handle. For this combination of reasons it becomes an optimum choice.

Calculations An equation of state (EOS) whether expressed in the usual variables P, V, T or in the more restricted set P, V, E used in hydrodynamics, can be thought of as a two-dimensional surface in the thermodynamic 3-space. A therlnodynamic constraint such as constant temperature or constant entropy can be thought of as an intersecting surface and will produce a line intersection which we will call a path. Such a path may be measured experimentally. The EOS surface can be generated in the neighborhood of the path if one of a class of thermodynamic derivatives can be obtained in that region. The class consists of those derivatives which lie in the EOS surface and which have a component which is not parallel to the path.

493

494

EQUATION OF STATE OF DETONATION PIIODUCTS

o.91

I

I

I

0.8

0.7

0.6 15" ~0.5

G = 0.45 [] - Our data 11 zx- Navord 1 o - Dremln

E

U

~0.4 C3

v - LASL 2

0.3

[ ] - Calculated G = G(V)

0.2

0.1

I

I

I

I

0

0.5

1.0

1.5

L O A D I N G DENSITY (g/cm 3) FIo. 1. Dependence of detonation velocity on loading density. For example, an equation of state can be generated from a known thermodynamic path such as an adiabat or isotherm and the specification of a derivative quantity such as (OP/OT)v or G=--V(OP/OE)v= (V/Cv)" (OP/OT)v. We have measured the CJ state and the adiabatic expansion of the detonations products of P E T N (p0= 1.77). This adiabat will be the basis path and will be described by the equations

Ps = A exp (-- RI" V) + B exp (-- R2* V) --f-C / V (1+~~

0.002 g/ee, e0=0.101 Mb co/co, e0 is the heat of detonation as measured in a detonation calorimeter. Correction is made for the standard state of water. We have used H20 (g) at 298~ and 1 arm. The Hugoniot relation is e--e0= 1/2 (P+Po)" (1--V). V is relative volume V/Vo. We have calculated6 the radius-time history in the cylinder test experiment to within 1% of the experimental data using this description. We estimate that the pressures above 1 kb are thus measured to within 2%. The equation of state in PVE variables is obtained directly from

Es = (A/R1) exp (--Rt. V) + (B/R2) exp (--R2. V)+ (Cflo)(1/V~)

(2)

a =_v (oP/OE)v dP = [G (V)/V-]dE,

with A=7.9723, B=0.19434, C=0.006006, and Rt=4.8, /72=1.2, ~=0.23. The CJ condition which we have recently measured is PcJ = 0.32+ 0.01%{b, D=0.83=1=0.002 cm/#sec, p0=1.77=t=

P= Pz+

/;

(3)

[G ( V ) / V ] dE

S

=Pz+[-G(V)/V-](E--Es).

(4)

DETONATION PRODUCT GASES The specification of G(V) is accomplished by using this equation of state to calculate the Chapman-Jouguet states for various loading densities with trial functions for G (V) and matching the calculated results to the measured detonation velocities as a function of loading density. The experimentally observed detonation velocity dependence was obtained from a summary of values in the literature supplemented by our data, and is illustrated in Fig. 1. These data give evidence of a decrease in slope at the higher densities. We have included this in the calculations and constrained the curve to match our data point at D = 0.830 cm/#sec. The value of G at the CJ point of the basis adiabat is calculated by using the Jones s relation, which can be expressed as

a0= r ( r - 1 - 2 z ) / ( v - z ) where

z= (p/D) (OD/Op). The detonation velocity has not been measured below po= 0.24 but can be estimated from thermodynamics. For low densities ideal gas relations should apply, F as used in this report is defined as F = - - (O l n P / O

lnV)z.

From thermodynamics --F

cp/c,= (c) lnP/O lnV)T"

0.9

1

0.8

495

Thus for ideal gases (PV/RT= 1 ),

c,,/co= r Since the high temperature value of Cp averaged for the various gaseous species is 10.7 cal/mole, F c j should be 1.23. Using this latter value for F c j and the Chapman-Jouguet relation, D2= 2E0 (F 2-- 1 ), E0 = 0.057-- (tcrergs/g), we find that D = 0.24 at po = 0. The values used at p=1.77 were dD/dp= 0.34-4-0.01, Z= 0.725-4-0.02, F = 2.812=1=0.06, giving G=0.48-4-0.10. The value of G at the CJ point of the basis adiabat is thus rather poorly defined. However, as we will show the dependence of detonation velocity on density is quite sensitive to G(V) as one proceeds to lower densities, and is therefore better defined on the rest of the curve. The increase in G as density increases is in the direction one would expect in approaching a "condensed" system where, in solids for instance, G is generally in the range 1 ( G ~ 2. The function G (V) which fits the data behaves as shown in Fig. 2. The empirical equation used for G (V) is given in the Appendix. The ChapmanJouguet state calculations were accomplished by iteration, minimizing the slope P/(Vcs-- V0) consistent with E--Eo=P/2(Vo--V) and Eq. (2). I t is assumed that E0 is a constant equal to 0.057 teraergs/g, i.e., (dEo/dVo)=O. This assumption has been examined for the case of P E T N by Wood and Fickett 9 who concluded that the variation is negligible. The calculated D versus p0 dependence is shown as the solid

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Fickett's LJD calculation

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1.0 1.5 GAS DENSITY (g/cm 3)

-

-

P 0 -- 1 . 7 3

-

P0 = 1.02

_

I 2.0

FIG. 2. Dependence of Graneisen's ratio on density.

2.5

496

EQUATION OF STATE OF DETONATION PRODUCTS

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F 2

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10

V/V 0 Fro. 3. F dependence on volume expansion for various loading densities; p0 equal to (1) 1.77, (2) 1.67, (3) 1.4, (4) 1.2, (5) 1.0, (6) 0.8, (7) 0.6, (8) 0.4, and (g) 0.3. line in Fig. 1. The same comparison for G= constant is also shown in Fig. 1 to indicate the sensitivity of the results to the value of G (V). Fickett 1~ has calculated thermodynamic parameters for P E T N detonation product adiabats using the same LJD equation of state as in Ref. 2. His calculations for P E T N at p0= 1.73 and po= 1.02 also shown in Fig. 2 predict somewhat larger values of G along the adiabat than our results would indicate, but tend to support our assumption (OG/Ot)v=O. Fickett's values for G (V) for the two adiabats very nearly coincide, in spite of the fact that the calculated temperatures differ by 800~176 depending on the gas density chosen for comparison. Fickett's calculation predicts the formation of some solid carbon in the high density region which casts some doubt on the calculated behavior of G in this region. The specification of G (V) allows us to predict the adiabatic expansion of P E T N detonated at any loading density. Since only the basic adiabat [Eq. (1)] is in closed form, we again have had to use iterative methods and a computer program to carry out the calculations. These adiabats are calculated by taking sufficiently small increments for AV and conserving the quantity.

(PAV+AE)= 0 which follows from the thermodynamic constraint

(OE/OV)s= - P .

The results, plotted as F versus V/Vo, are shown in Fig. 3. For comparison, another set based on G=0.45 which is a median value for G(V) is shown in Fig. 4. For a hypothetical experiment at po= 1.0, we have calculated the internal energy change during expansion for the two cases. Throughout the expansion there is a difference of 16% in energy expended. This would be detected as an 8% difference in wall velocity in a cylinder expansion experiment. G, as calculated by Fickett 1~ will produce an even larger difference. By performing an equilibrium calculation at larger expansions where the products are nearly ideal gases, one can obtain the temperature for specified values of P and V; then from thermodynamics,

--[G(V)/V]dV=dT/T

at constant S.

Therefore,

TiT1 = exp

U ( V ) / V dV

which can be integrated numericalIy. Thus we can calculate T at any point on the isentrope. The greatest uncertainty is the value of the adiabat pressures at large expansions. By varying the equation-of-state parameters, we find the matching pressures are determined to within

D E T O N A T I O N P R O D U C T GASES 3

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v,/v o FIG. 4. s versus V/Vo for G = G(V), and G = 0.45 at loading density po = 1.0. about 30%. The inatching point was chosen at 140 atm and V/Vo = 22 on the adiabat, Eq. (2). We used Fickett's 1~equilibrium calculations since in this region P V / R T "~ 1.00, and applied ideal gas corrections to his point at P = 300 arm, V/Vo = 19.4, T = 1235~ This gave 680~ as the base point for the integration.

Experimental Comparison I n the preceding section we have shown how we can calculate the C J pressure, temperature, and adiabatic expansion for any loading density, p0. In Table I we have listed predicted Pc~ and Tcz for 0.8 < p0 < 1.77. The data so far avail-

TABLE I CJ Parameters Experimental Density ( g / c m 3)* 1.77 (LRL) 1.671 (2) 1.66 (1) 1.51 (1) 0.95 (1) 1.00 (11) 1.2 (11) 1.4 (11) 1.6 (11)

D (cm/~sec) 0.8314-0.005 0.7974 0.81 0.742 0.530 0.555 0.634 0.713 0.792

P (mbar) 0.3204-0.005 0.300• 0.246 0.187 0.0645

* References are in parentheses. t Author suggests pressure may be higher.

T (~ 4200 (12)* 3400•

Calculated D

P

0.830 0.798 0.794 0.743 0.535 0.555 0.631 0.704 0.774

0.320 0.279 0.274 0.215 0.0726 0.0815 0.124 0.178 0.248

T (~ 2070 2340

4300 3580 3270 2530

498

EQUATION OF STATE OF DETONATION PRODUCTS TABLE II

tions. The gas density p (g/cc) is the independent variable.

Coefficients for G(V) 1

2

3

0.11 --0.21 0.09 0.40

15.0 4.5 7.5 --

1.35 0.71 0.25

G(V) = A, tanh A2[A3 -- (p/1.77)] +B~ tanh B2[B~ -- (p/1.77) ]

A B C D

--

able are limited to a few Chapman-Jouget pressure measurements and two temperature measm'ements which are compared with our predicted values in Table I. Our Tcj result is surprisingly low at p0 = 1.77 and 1.67, but we find that the parameter changes necessary to produce temperatures as high as those reported cannot be justified.

Conclusion

Since, unfortunately, there is considerable disagreement among the sources of PcJ data and no adiabat measurements have been made at lower densities, further work will be necessary to test the consistency of our analysis. Since any large deviation from consistency will require re-examination of the basic assumptions used in the analysis, we feel such experiments could be an important contribution. Some of these experiments are now planned or are in progress at our Laboratory.

+C~/cosh C2[C3 -- (p/1.77)~ + D~. The tanh function behaves as a modified step function, and the reciprocal cosh function behaves as a modified delta function. Initial guesses can be made quite easily as compared with polynomial expansions of equal complexity. The values of the constants are given in Table II.

REFERENCES 1. DRE~IIN,A. N. ANDSHVEDOV,K. K.: Zh. Prikl. Mekh. Tekn. Fiz., 3, 154 (1964) (Translation). 2. FICKETT, W.: Detonation Properties of Condensed Explosives Calculated with an Equation of State Based on Intermolecular Potentials, Los Alamos Scientific Laboratory Report, LA-2712, 1962, p. 134. 3. JACOES, S. J.: ARS J. 30, 151 (1960). 4. FICKETT, W. AND WOOD, W. W. : Phys. Fluids 1, 528 (1958). 5. DEAL, W. E.: Phys. Fluids 1, 523 (1958). 6. KURY, J. W., ItORNIG, H. C., LEE, E. L., MCDONNEL, J. L., ORNELLAS, D. L., FINGER, M., STRANGE, F. M., AND WILKINS, M. L.: F o u r t h

Symposium on Detonation, p. 3. ACR 126, ONR (1965). 7. ORNELL&S, D., CARPENTER, J. H., AND GUNN,

ACKNOWLEDGMENTS

We thank John Cast and Margaret Jepson for assistance in carrying out PcJ and detonation velocity experiments. This work was performed under the auspices of the U.S. Atomic Energy Commission.

S. R.: Rev. Sci. Instr. 37, 907 (1966). 8. JONES, H.: Third Symposium on Combustion, Flame and Explosion Phenomena, p. 590, Williams and Witkins, 1949. 9. WOOD, W. W. AND FICKETT, W. : Phys. Fluids 6, 648 (1963). 10. FICKETT, W.: private communication. l 1. CHRISTIAN, E. A. AND SNAY, H. G. :

Analysisof

Experimental Data on Detonation Velocities, NAVORD Report 1508, 1956.

Appendix

The function G ( V ) illustrated in Fig. 2 was calculated in closed form using hyperbolic func-

12. APIN, Y. A. AND VOSKOBOYNIKOV, I. M.: HMFT, 117, No. 5, (1961).

COMMENTS L. B. Seely, Stanford Research Institute, Menlo Park, Calif. Is it possible that solid carbon is formed at high density in PETN, and might this explain the bending-over of the D(p) curve? You have quoted experimental temperatures. Who measured them and by what method?

E. L. Lee. Although some calculations, notably W. Fickett's (Los Alamos Scientific Laboratory, Los Alamos, New Mexico) calculation, indicate carbon formation, there is some evidence in Hayes's measurement of conductivity behind the shock front that the amount of carbon formed in P E T N is very small, if present at all. Thus, the

DETONATION PRODUCT GASES "bending over" may more likely be due to properties of the fluid. At p0 = 1.67, the measurement was made by W. Davis at LASL, and at p0 = 1.77 by Y. A. Apin, A M F T , 117, No. 5 (1961). Fickett listed W. C. Davis's value in his report LA-2712. The method in both was photometric, but I would not comment further on these measurements. Most people seem to consider them very doubtful.

A. N. Dremin, Institute of Chemical Physics, Moscow. I consider that all data on detonation temperatures, based on any measurements of luminescence of detonation products or detonation fronts, are doubtful, because we cannot be sure that this luminosity originates at the C - J plane.

C. L. Mader, Los Alamos Scientific Laboratory, Los Alamos, New Mexico. All the experiments that can be performed with explosives are somewhat divergent and somewhat time-dependent. All the theory for the propagation of detonation is for steady, one-dimensional flow. In the comparison of experiment with computation to determine the equation of state of an explosive, the degree of approximation present in the theory behind the computational model is of vital interest when one wants to know how closely the apparent equation of state approximates the actual equation of state. A theoretical treatment of a spherically expanding detonation wave does not exist. The Taylor self-similar solution for divergent detonations has been widely used because a better treatment does not exist; however, Courant and Friedrichs~ show that it is not correct. The Taylor self-similar solution does not permit the pressure at the end of the reaction zone to change with the divergence of the flow. While a theoretical treatment does not exist, we have used a computational model that appears to be realistic, since it reproduces the available one-dimensional divergent flow experimental data. The results from this model are that the pressures at the end of the reaction zone are 25 per cent lower than one would obtain using the Taylor self-similar solution for the spherical geometries used to initially

499

calibrate the equation of state used by Lee and Hornig. Experiments show that the detonation front in a cylindrical charge is appreciably curved rather than plane as assumed by Lee and Hornig. I t is difficult to estimate the degree of approximation introduced by this assumption. The time-dependent features of the detonation process are also important. Some experiments have shown relatively large effects even in quite large charges, but the details remain unclear. If one uses the infinite medium, steady-state pressure at the end of the reaction zone (which is too high because the divergence and timedependent features are not included), it will be necessary to have the steep isentropes reported by these authors to reproduce the gross observed experimental behavior. The intent of these remarks is first to encourage more work on theoretical and computational models that describe the real experimental arrangements, and second to question whether the equation of state derived from experiment and analyzed in the customary manner is a real equation of state. REFERENCE 1. COURANT,n. AND FRIEDRICHS, K. O.: Supersonic Flow and Shock Waves, p. 430, Interscience, 1948.

E. L. Lee. The crux of this criticism is contained ir~ the statement "Experiments show that the detonation front in a cylindrical charge is appreciably curved rather than plane as assumed by Lee and Hornig. I t is difficult to estimate the degree of approximations introduced by this assumptions." The point we should like to make is that we did not assume this, but carried out scaling experiments in 1", 2", and 4" cylinders and multislit experiments in which we observed the motion of a 1" cylinder over a range of LID from 1.510.5. This was to assure ourselves that selfsimilar flow was attained. Thus, although we are perfectly aware that non-self-similar flow can be observed in many experiments (diameter effect, for instance), in our experiments, this is not the case, and our equation-of-state results should not be affected by this difficulty.