M. A. COOK,R. T. KEYES,G. S. HORSLEY AND A. S. FILLER
1114
However, a t the point of contact (which occurred a t the plait point of the bite) the border of the band just before contact was distinctly concave. This is a theoretical requirement to satisfy Schreinemakers’ r ~ l e with ‘ ~ ~respect ~ ~ t o triangular areas representing three phases. That rule would prohibit the more usual illustration of merger of convex curves.20*21 The three-liquid phases in some of the carbon
Vol. 58
dioxide systems result from the above mechanism. Those of the others, including the formic acid systems mentioned above, and probably all of the published systems with three liquid phasesz2result from eruption of a second binodal curve from within another one a t a point other than the latter’s plait point. This is possible, and is the type cited by Hillz1in support of his hypothetical diagram. (22) Reference 5 , pp. 847, 977, 1009, 1015, 1029-31, 1035-6, 1070.
A STUDY OF THE EQUATION OF STATE FOR EDNA1 BY MELVIN A. COOK,ROBERTT. KEYES,G. SMOOT HORSLEY AND AARONS. FILLER Explosives Research Group, University of Utah, Salt Lake City, Utah Received J u n e 1 1864 ~
Therinoliydrodpamic calculations were made by the “inverse” method (measured detonation velocity-density equation included in the solution) for EDNA using three fundamentally different equations of state leading to widely different internal pressures ((dE/dv)T). TWO sets of measured velocity data were used with each equation of state. The results show that all the calculated thermodynamic quantities except temperature are less sensitive to the form of the equation of state than to errors in the determination of detonation velocity. Hence, temperature alone provides an adequate criterion of an objective evaluation of the equations of state when one approaches the problem sole1 from detonation theory. However, so far reliable detonation temperature measurements have not been possible, and t h s criterion cannot therefore be applied. A corollary of this conclusion is that any reasonable equation of state provides, through detonation theory and measured velocities, as reliable thermodynamic data as any other. Objective detonation equation of state studies must evidently await more accurate velocity-density measurements and the development of methods for measuring some detonation property with sufficient accuracy to allow one to evaluate the various forms of the equations of state unambiguously.
Numerous equations of state of various forms have been used in thermohydrodynamic calculations. In spite of wide differences in form and character of these equations of state the thermodynamic quantities computed, either by direct use of experimental detonation velocity us. density data or by adjustment of parameters to give best agreement with observed velocities, have been in surprisingly close agreement, except for computed detonation temperatures. Temperature alone turns out to be strongly dependent upon the nature of the equation of state. This situation led one of us2 to conclude that the detonation temperature is the only factor where a comparison between computed and observed values could be used to evaluate the accuracy of the equation of state. Unfortunately, even such a comparison with detonation temperatures is inadequate in view of the great limitations of temperature measurements. It was therefore considered advisable to make a thorough theoretical study of the influence of the form of the equation of state on the various thermodynamic quantities computed from the thermohydrodynamic theory and also to study theoretically the influence of experimental errors in velocity The explosive EDNA (Haleite) a was selected for this study since it appeared to be well suited both from the viewpoint of reliability of computed products of detonation and from measured velocities. In fact, two sets of velocities have been obtained showing good agreement at high densities but differing considerably in the velocity a t low
.
(1) This project was supported by Office of Naval Research (Contract Number N7-onr-45107, Project Number 357 239). (2) M. A. Cook, J . Ckem. Phus., 16, 518 (1947). (3) Ethylenedinitramine OzNHNCHzCHzNHNOn.
density and in the slope of the velocity-density curve. These were as follows4 D = 5650 386O(pi - 1.0) (la) D 5960 3275(pi - 1.0) (1b) =i
+ +
(See Appendix I for definitions of symbols)
The general equation of state p v = nRT’p
(2)
was adopted for this study, Specific forms of ‘p were selected such as to exaggerate differences in the equation of state, using the specific definition ‘p
=
e’
(3)
where TC
x = K ( v )V
Three cases were treated using the following values of c c = -0.25
c=o c = +0.1
(3a) (3b) (3c)
Definition 3a leads to an equation of state of much the same form as that of Kistiakowsky-WilsonBrinkley15although K is here allowed to vary with density (experimental velocities being used to determine K ) whereas in the KWB equation it is a constant. Also here ‘p = e x , but in the KWB case cp = 1 xex. One will, however, note that these two forms are not radically different. Definition 3b is equivalent to the a(v) approximation used by Cook2 and in different form by Caldirolaea and Paterson.6b Definition 3c is probably completely
+
(4) Measured a t Bruceton, Pennsylvania (NDRC, Division 8 ) . (5) OSRD No. 89, 905, 1231, 1510, 1707, NDRC Division 8 Staff. (6) (a) P. Caldirola, J. Chem. Phya., 14, 738 (1946); (b) 9. Patterson, Research. 1, 221 (1948).
EQITATION OF STATEFOR ETHYLENEDINITRAMINE
Dee., 1954
erroneous since it leads to an internal pressure which apparently is inconsistent with detonation conditions. Derivation of Equations and Methods of Solution. -The fundamental equations of the hydrodynamic theory are : (Conservation laws)
Using equation of state 2 (with
p =
ex, and x
1116
this problem it is suggested that perhaps the most efficient method in many cases of providing the necessary product compositions is to compute for the explosive a series of composition us. fugacity isotherms covering a broad enough range to include the real solution in order to have the necessary composition data available for use in the much easier thermohydrodynamic calculations. This more systematic method will here be illustrated for EDNA. Defining the equilibrium constants Ki in the manner suggested by Brown,' using the equation of state given by equations 2 and 3 following the recent methods,8one obtains
=
K(v)Tcu-l) one obtains the relations 0 2
= nRTze"(1
+ O)z/e
(5)
E,FA"
where
(11)
j
( T z = constant)
where F
= 'p
exp
(2'2
Jp(p
=
-
I ) d In p
(12)
constant)
Introducing equation 3, equation 12 becomes ( T I = constant)
(7'2
constant)
Figures 1 to 5 give the various isotherms needed to carry out the thermohydrodynamic solution by W 2 = nRT2(9) the simultaneous solutions of equations 5 to 12. e Incidentally, one will note from Fig. 1 that the Tz = (Q C,TI)/(C, - n ~ e z / 2 0 ) (IO) equilibrium CO 2H2 Fs, CH30H not previously (Derivations are given in Appendix 11.) considered was found to be important a t large F. The K(v) function can be evaluated, in lieu of This will be discussed in a future article where the any fundamental basis for it, only by introducing necessary heat and statistical mechanical data to the experimental D(pl) data given by equation 1. justify the use of this and other previously ignored I n solving these equations one needs also to know possible products of detonation and their influence the composition of the products of detonation for in determining over-all composition of detonation the particular conditions corresponding to detona- products will be discussed. I n the first approximation, one may neglect tion. In studies of detonation it is necessary to solve a dK/dvz and the integrals in equations 7, 8 and relatively large number of equilibria simultaneously 12a, and ignore the variation of n with F shown in for various specific conditions. The problem is Fig. 2. It is convenient to solve the equations for complicated and arduous since iteration of the various values of z (or p) noting that the term in numerous generally non-linear simultaneous equa- the exponential of equation 12%is then completely tions is required. Moreover, in the completion of determined (see Table I). The next approximaa single problem, i.e., a single explosive calcula- tion can now make use of these first approximation tion, solutions of the problem for many different plots of K(v) us. v2 and n(v) us. us allowing one to sets of conditions are required. For instance, in determine as a next approximation the integrals the problem of obtaining reliable thermodynamic and dK/dvz neglected in the first approximation. data from the hydrodynamic theory by the "in- This process is then repeated as often as necessary verse" method, one requires a minimum of about for complete stabilization of the approximation one hundred separate solutions each involving an variables. The solution in which one takes into iterative solution of about twenty-five simultane- account the factor (dn/dTz), in 8 is actually nonous equations. Furthermore, in many specific tractable. However, this is evidently not serious applications, numerous additional solutions are since one can show that the term involving this required for the various specific conditions for (7) F. K.Brown, Bureau of hlines Teolinicnl Paper No. G32. which data are desired. After extensive study of (8) M. A. Cook, J . Chem. Phya.. 16, 1081 (1948). (2'2
= constant) ez
+
.
=
+
M. A. COOK,R. T. KEYES,G. S. HORSLEY AND A. S. FILLER
1116
Vol. 58
14
-l
Ok-
.. I
SUBSTANCE
31
&-
NH, CH,-
--
2
I
00
1.0
2 .o
Fig. 1.-Concentration
3.0
4.0
5 .O
6 .O
LOG F.
versus F isotherma for EDNA.
factor, a t least in the case of EDNA, is negligible, e.g., a t F = lo4 and 35OO0K., T2/n (dn/dTz),. is about 0.05 which is its maximum value. If (dn/dTz). mere not negligible, one could not of course drop the subscript S in the factor (dn/dvz)s. In this study, however, (dn/dT2), was taken as zero, and dn/dvz evaluated from successive plots of n us. v2. I n one set of calculations (dn/dvz)s was neglected as has been done in all previous theoretical treatments, i e . , in all previous work
n has (tacitly) been taken to be constant. The influence of treating n as a constant as compared with the assumption n = n(v) is illustrated in the calculated results of this article. The 'integrals in equations 7, 8 and 1% should be evaluated a t constant temperature. Actually they were computed a t the calculated detonation temperatures. The error so introduced is small for the equation of state determined by (3a) since then the detonation temperature is almost con-
*
EQUATION OF STATE FOR ETHYLENEDINITRAMINE
Dec., 1954
TEMPERATURE
-
46
1117
2500' 3000' 3500' 0 4000' x 4500' 5000' a 5500" 0
a
-
44
7
-'s42 c
a
8
sZ 40 C
v)
w
-
- 38
-
-1
i
-
36 -
I
32 1
ao
__
1
I
IO
20
I.-
30
40
50
60
LOG F ,
moles of gas per kilogram versus F isotherms.
Fig. 2.-Total
L
1220-
TEMPERATURE 0 2500' * 3000'
0
1180-
x
-
7
-.
-
3500' 4000' 4500' 5000O 5500"
I 140-
Y CII
3 1100-
Y
/
1
/////
1020980I
0.o
I
1.0
3.O
2.0
4.O
5.0
60'
LOG F.
Fig. 3.-Heat
of explosion versus P isotherms.
stant. I n the case of the equation of state determined by (3b), since the integrals in (7) and (8) are multiplied by c, which is zero, the only error is in (12a). I n the case of the equation of state determined by (3c) there will be error in all three equations 7, 8 and 12a. One of the advantages of the type of equation of state given by equation 3 is that the major parts of the equations 7 , 8 and 12a are given in analytical form. The analytical parts are more important than the parts given by the integral, and the effects of errors in the integrals are thus reduced.
Results of Calculations Calculations were first carried through with the three equations of state defined by equations 2 and 3 for the velocity data given by equation la. I n these calculations the n = n(v) approximation was employed. T o show the influence of this more refined treatment, calculations were also carried through for equation of state (3b), pv = nRTeK(QV, treating n as a constant. Then to show the influence of measured velocities on these calculations, results were repeated with two of the equations of state (z defined by 3a and 3b) and the set of
1118
M. A. COOK,R. T. KEYES,G. S. HORSLEY AND A. S. FILLER
Vol. 58
48,
1
TEMPERATURE
46 -744 0
-
Y
1
2500' 3000' 0 3500' 0 4000' * 4500' 0
E.
5000'
5500"
y"42 -
-
s
-
Y v
340 38
36
-
1
,
1
EQUATION OF STATE FOR ETHYLENEDINITRAMINE
Dee., 195-4
,
.42
1119
_ I _ -
A0 -
c.38
-
0I
Y
-
-
.E.36
-8
-
-
Y
*>34 '0
32 30 0.0
I .o
2.0
3.0
5.0
4.0
6.0
LOG F.
Fig. 5.-Average
heat capacity (between 300°K. and T ) versus F isotherms.
different velocities one calculates pressures differing by as much as 30y0 (at the density of maximum difference). The calculated temperature is enough more sensitive to the form of the equation of state than to experimental errors in velocity that it could readily be used to ascertain the correct equation of state if it could be measured. Unfortunately, this has not yet been possible as far as the detonation temperature Tz is concerned. From the results of this study, it may he concluded that fundamental theoretical-experimental evaluations of the equation of state applicable in detonation must await (1) reliable and accurate measurements of temperature, and/or (2) the I293
developments of techniques which will allow one to determine detonation velocity us. density dependent parameters not only a t high densities] but also a t low densities with an experimental error of 0.5% or less, together with experimental techniques for the measurement of pressure, composition, density, and/or particle velocity with suitable precision. For instance, if it were possible to measure detonation pressures within 2% and velocities within 0.5%, one could distinguish satisfactorily between the three equations used here. Available techniques for velocity determinations are already adequate, the limitation being wholly in obtaining reproducible densities a t low density,
1500
610
-----1207
C.-0.25
c -0.10 .. ............... , . '"C.0,
% TERM
1400
1121
*-"
I
D=565O+ 3860(
