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The properties of gases at high pressures which can be deduced from explosion experiments
590
THIRD SYMPOSIUSI ON COMBUSTION~ FLAME AND EXPLOSION PHENOMENA
6. BECKER, R.: Z. Elektrochem., 23, 40, 304 (1917); Z. Phys., 8, 321 (1922) 7. JOST, W.: Z. Elektrochem., 41, 183 (1935); Z. Phys. Chem., 42, 136 (1939). 8. SCORAH,R. L.: J. Chem. Phys., 3, 425 (1935). 9. KISTIAKOWSKY,G. B., AND WILSON-, E. B., JR.: OSRD Rep. 114 (available from Off. Tech. Serv., U. S. Dept. Commerce, Report PBL 32715). See also the summary by H. L. Dryden, F. D. Murnaghan, and H. Bateman, Bull. Nat. Res. Council, No. 84, 1932, p. 551. 10. BRINKLEY,S. R. JR., AND KIRKWOOD,J. G.: Phys. Rev., 71, 606 (1947).
identification of the detonation wave in an actual explosive with the model that has been employed in the present discussion. REFERENCES 1. RANKINE, W. J. M.: Trans. Roy. Soc. London,
A 160, 277 (1870). 2. HUGONIOT, H.: J. de l'6cole polyt., 57, 3, 1887; 5,S', 1 (1888). 3. CHM'MAN,D. L.: Phil. Mag., (5), 47, 90 (1889). 4. JOUGtTET,E.: Comptes Rendus, 132, 573 (1901). 5. LEWIS, B., AND FRIAr:F, J. B.: J. Am. Chem. Soc., 52, 3905 (1930).
76
T H E P R O P E R T I E S OF G A S E S AT H I G H P R E S S U R E S W H I C H CAN BE D E D U C E D F R O M E X P L O S I O N E X P E R I M E N T S ~ By H. JONES INTRODUCTION
Following the successful theory of the detonation of gases ~ontained in tubes formulated by D. Chapman (1) and E. Jouguet (2), at the beginning of the century, attempts have been made, at first notably by Schmidt (3), to apply this theory to solid explosives. In order to determine the velocity of the detonation wave theoretically a knowledge of the equation of state of the product gases of the explosion is required in a range of temperature and pressure far beyond those realizable in ordinary laboratory work. The pressures involved are of the order l0 s atmospheres and temperatures of the order of 3000~ Since the detonation velocity is a function of the loading density of the explosive, and since these velocities can be measured with considerable accuracy, it is natural that attempts should have been made to reverse the argument and to deduce information about the equation of state from the observed detonation rates. I t is the purpose of this paper to discuss the extent of the information concerning the equation of state which can be obtained in this way from detonation velocities. Care has to be taken when applying the ChapThe author is indebted to the Director-General of Scientific Research (Defence), Ministry of Supply, for permission to publish thls paper.
man-Jouguet theory to rods of solid explosives because this theory applies only to a strictly one dimensional detonation wave. It has been shown (4) that the lateral expansion in the reaction zone of a detonation wave causes a diminution in the velocity, which is given by an expression of the type
(Uo/U)' = 1 + f(l/r)
(1)
where f is a function which can be calculated approximately and which diminishes rapidly as l/r diminishes. U is the actual detonation velocity and U0 that which would be given by a correct application of the Chapman-Jouguet theory, r is the radius of the charge and l the length of the reaction zone. Thus the only data which should be used in an attempt to obtain information about the equation of state are velocities determined for charges of diameter large enough to give the maximum detonation rate. For most high explosives, charges of diameter of about one inch usually give the maximum velocity, but for the low power explosives, e.g. those containing much ammonium nitrate, the observed velocity will not at all correspond to the limiting value, which alone can be used with the Chapman-Jouguet theory to yield information on the equation of state.
B U R N I N G A N D D E T O N A T I O N OF E X P L O S I V E S
and
w THE PRESSURE AND DENSITY IN THE DETONATION f R O N T
OF l a G _ OF / O G
Let p denote the pressure and v the specific volume of the gaseous products in the detonation front, and let E(p, v) be the internal energy of the gas per unit mass measured relative to the energy of the unexploded material at atmospheric pressure, and Q be the chemical energy liberated per unit-mass by the reaction which, in general, is also a function of p and v. The well known equations of the Chapman-Jouguet theory can thus be written
E(p, v) - O = 89 p =
V2p(1
-
5Q1
- v)
(2)
pb),
(3)
where U is the detonation velocity and p the loading density of the explosive. The values of p and v in the detonation front are given, according to the Chapman-Jouguct theory, by the point in the p, v plane where, for given U and p, the straight line (3) touches the curve (2). Instead of following the usual procedure of assuming a form for the equation of state and thereby deducing p and v and hence U for each value of p and comparing this calculated dependence of U on p with experiment, we regard U as a known function of p and from this obtain directly information about the equation of state. Regarding p, v and p as independent variables equation 3 determines a ruled surface which touches the curved surface (2) along a certain line. According to the Chapman-Jouguet theory the values of p and v along the line of contact determines the pressure and specific volume in the detonation front for any given value of the loading density p. The condition that the surfaces (2) and (3) touch leads to two equations which express the fact that at any point of contact the direction cosines of the normal to each surface are equal. It is convenient to write 1/p = z, and g(z) = U2p2 which is to be regarded as a function of z, and then to express equations 2 and 3 in the form
F(v, p, z) =-E (p, v) -- O(p, v) -- 89p(z -- v) = 0
(4)
G(v, pm z) = g(z)(z - v) - p = 0
(5)
bz / oz
(7)
Ov /
Equation 6, together with equations 4 and 5, gives
0 Or ( E -
Q) + 8 9
pp ( 0 1 -pv.~(E-Q)
which together with equation 4 determines p and v when E and O are known. Equation 3 then determines the detonation velocity for any given p. This leads, therefore, to the usual ChapmanJouguet theory. Equation 7, on the other hand, leads to a result which does not appear to be well known. From equations 4, 5, and 7, we find
89
T,,
9(1 +
-
~ ]
(9)
Writing (6/~)v)(E - Q) = (1/~)p, which defines the quantity a, equation 9 together with equation 5 gives easily
U=p P =
{ (2+a)
l+d
d log U \ log p j
(10)
and
=
1(2 +
1+
dlogU (11) log p)_l
Thus apart from a, which we discuss below, these equations determine the pressure and the specific volume in the detonation front, from the observed U, p relation, without any assumption concerning the equation of state. Another result of the Chapman-Jouguet theory is that the fluid velocity, relative to the detonation wave, is just equal to the local velocity of sound. In symbols this relation is
The conditions for contact are then given by --V2(bp /Dv), = p2v~U~
OF /
OF _ 0(7 /
a;/oi,
OO
o,/op
(12)
(6) where (i~p/Sv), denotes the adiab~,tic variation of
592 pwithv.
T H I R D SYMPOSIUM ON COMBUSTION~ ]FLAME AND E X P L O S I O N P I t E N O M E N A
Thus if "r =
\ 0 log v/~
(13)
so that y is the initial value of the exponent in the adiabatic relation/~'2 = const., then we have from equations 10, 11 and 12
= (2 +
1+
d log U l
ag-og, J -
1.
(14)
We now show that a is essentially positive and therefore equation 10 gives an upper limit to p with a = 0, and similarly equations 11 and 14 give lower limits to v and ~,. The quantity E - O is the internal energy of the gaseous products relative to a standard state in which the constituents are taken as elements in their normal state. Thus since
it is clear that c~ is positive, for both the energy and the volume are increasing functions of T at constant pressure. The specific heat at constant pressure for the gas mixture is given by
Hence (15) and (16) together with the equation defining a give a={-~(~-v)
- - 1 } -1.
(17)
Equation 13 together with the well knov~n thermodynamic relation for the ratio of the specific heats gives C,(0T)
=
7C,,(Oj)
(18)
x~.here C~ is the specific heat at constant volume. Thus we can put finally
a = [y,(Op/OT).
1
(19)
a is, of course, a function of p and v and therefore equations 10 and 11 do not give p and v explicitly; the usefulness of this form is due to the fact that 0 < a < < 2 and therefore even without any
knowledge of the equation of state we obtain approximate formulae for p and v. Using equation 19 the value of ~ can be obtained approximately, since the specific heat at constant volume can be rather accurately estimated from the well known specific heats of the product gases at high temperatures. An uncertainty arises on account of the high pressure. The effect of pressure, however, can only increase Cv, hence omitting the effect of high pressure will tend to over estimate a. To estimate v(i)p/bT)~ the equation of state deduced by Becker (5) for nitrogen was used. From these estimates it becomes clear that the least value which is at all likely for .rC~,/(Opv/~)T)~ is about 5 so that = is not more than 0.25. Thus it can be seen that, if the observed relation between U and p is used, it is possible to find rather accurate and reliable values of the pressure and density in the detonation wave front as well as the initial value of the exponent for adiabatic expansion, in spite of the lack of information about the equation of state in the range of p and v which exists in the detonation wave. With regard to the temperature of the detonation front, it is clear that the U, o relation cannot determine it since only the variables p and v occur in the fundamental equations. The temperature can, however, also be calculated with fair accuracy and reliability because it is largely determined by the specific heats of the product gases, which for low pressures are well known, and by the heat of the reaction Q. However, one cannot determine theoretically the composition of the products without knowing the equation of state, so that an uncertainty arises in this respect. Also one cannot infer the composition of the products from closed bomb experiments, since the products after adiabatic expansion will, in general, be different from those in the detonation front since chemical reactions will proceed during the expansion. However, as a rule fairly reliable values of T can be obtained which are not very sensitive to the assumed form of the equation of state. I t is now possible to appreciate the limited information which can be deduced from an observed U, o relation. We can expect to obtain a set of p, v, T values, for the products of detonation, over a certain limited range of a single variable. The equation of state of the gas mixture may be expressed as
f(p,
v, 2", N a , 2VB, - . . ) = 0
(20)
where the N a etc. are the mole fractions of the gas
BURNING
AND DETONATION
mixture. When the NA" 9 9 are held constant the equation 20 represents a surface in the p, v, T space. The maximum amount of information which can be derived from a U, p relation would fix a line of a certain length in this surface. Thus if one attempted to analyse the detonation velocity data by first assuming an equation of state, and allowed this equation to contain a large number of undetermined parameters which could, in principle, be evaluated from the U, p relation, yet the resulting equation of state would be very largely arbitrary since there are evidently any number of surfaces which contain the same line. The conclusion is that very little information about the form of the equation of state can be obtained from detonation velocity data but that if the form is assumed then the values of parameters in the equation can, in general, be determined. w
593
OF EXPLOSIVES
" 4, ~. ~, ~. &2
o
,
z
~
r
.
i ~,,o* i$
Fro. 1. Curve (a) determined by the detonation velocity-loadlng density relation. Curve (b), BridgII2an~s measurements on nitrogen.
f'*
1.2
N U M E R I C A L R E S U L T S I N T H E CASE OF P E N T H R I T E
As an illustration of the kind of information which can be obtained with the assumption of a particular form of the equation of state, this section gives the results of a calculation based on the U, p data published by Friedrich (6) for the explosive penthrite CsHsOI2N4. The products of detonation may be assumed to consist of CO~, CO, H~O, H2, N2; the relative amounts being determined by the assumption of thermal equilibrium3 Whatever the relative amounts of these gases may be, one gm. moh of penthrite will always yield 11 moles of the gas mixture, A very simple form for the equation of state of the gas mixture was chosen, viz.
pv = R T + f(p)
(21)
w h e r e f is some unspecified function of the pressure alone which is to be determined from the U, p relation, R is the gas constant per gin. of the gas mixture. The total energy of the gas, allowing for the variation of the composition, was calculated as a function of T u s i n g specific heats determined from spectroscopic data, and was found to be very nearly linear with T in the range 2000 ~ to 4000~ When the equation of state is of the type (19) and the energy is linear in T, ~ becomes a constant, . and in the present case was found to have a value
,?, ,.0 O
~ ae
> c~ ~o 0.4-
a2
to
20
5o
&o
5o
Fro. 2. Potential energy of molecular repulsive forces. 0.200. Hence the pressure in the detonation front, as a function of loading density, is obtained immediately from equation 10 and a simple calculation shows that f(p) is given by the following formulae: Ot
f(p) = r
_(a/l+a)
+ ~,p op p-(1+2,~/l+,,)g, dp
2 Calculations of this type are given in detail in a forthcoming paper to appear in the Proc. Roy. Soc. by H. Jones and A. R. Miller.
6o
CUBIC A.U. PER MOLECULE
(22)
where ~b is a function of p given in terms of the
594
T H I R D SYMPOSIUM ON COMBUSTION~ F L A M E AND E X P L O S I O N P H E N O M E N A
parameter p by equation 10, and (1-4-a)r
v2 2(2+a)
( 1-t-2d
dl~
9
1
-4- 6 1 ~ , /
dlog U~ logo,/ (23)
where Q is the constant heat of reaction. Using the experimental data given by Friederich (6) (1933) the values of (pv - RT) have been calcula[ed as a function of p for pressures up to 280,000 atms. Figure 1 shows the results together with Bridgman's (7) observed values for N= at 68~ up to 15,000 atm. An equation of state of the type (21) leads to a term in the internal energy of the form
which is the energy arising from the repulsive forces between the molecules. Figure 2 shows this energy as a function of the molecular volume.
If the equation of state, thus obtained, is now used to determine the adiabatic law and hence to obtain the initial shock wave velocity from the end of a bare charge, very good agreement is obtained with experimental values. This, however, is again no justification for the assumed form of the equation of state because the iDitial shock wave velocity is determined mainly by just those parts of the p, v, T surface which are determined by the observed U, p relation. REFERENCES
1. CHAPMAN,D. L.: Phil. Mag, 47, 90 (1899). 2. JOUOUET,E.: J. Maths. pures appl., 6 (Series II), 5 (1906). 3. Scrr~io'r, A.: Zeit. ges. Schiess-Springstoffw.,31, 8, 37, 80, 114 (1936). 4. JONES, H.: Proc. Roy. Soc., A189, 415 (1946). 5. BEC~ER, R.: Zeit. fur Physik, 4, 393 (1921). 6. FRIEDERmH, W.: Zeit. ges. Schiess-Sprengstoffr 28 (1933). 7. B~DG~N, P. W.: Proc. Amer. Acad., 49 (1913); 9 (1923); 59 (1923).
THIRD SYMPOSIUSI ON COMBUSTION~ FLAME AND EXPLOSION PHENOMENA
6. BECKER, R.: Z. Elektrochem., 23, 40, 304 (1917); Z. Phys., 8, 321 (1922) 7. JOST, W.: Z. Elektrochem., 41, 183 (1935); Z. Phys. Chem., 42, 136 (1939). 8. SCORAH,R. L.: J. Chem. Phys., 3, 425 (1935). 9. KISTIAKOWSKY,G. B., AND WILSON-, E. B., JR.: OSRD Rep. 114 (available from Off. Tech. Serv., U. S. Dept. Commerce, Report PBL 32715). See also the summary by H. L. Dryden, F. D. Murnaghan, and H. Bateman, Bull. Nat. Res. Council, No. 84, 1932, p. 551. 10. BRINKLEY,S. R. JR., AND KIRKWOOD,J. G.: Phys. Rev., 71, 606 (1947).
identification of the detonation wave in an actual explosive with the model that has been employed in the present discussion. REFERENCES 1. RANKINE, W. J. M.: Trans. Roy. Soc. London,
A 160, 277 (1870). 2. HUGONIOT, H.: J. de l'6cole polyt., 57, 3, 1887; 5,S', 1 (1888). 3. CHM'MAN,D. L.: Phil. Mag., (5), 47, 90 (1889). 4. JOUGtTET,E.: Comptes Rendus, 132, 573 (1901). 5. LEWIS, B., AND FRIAr:F, J. B.: J. Am. Chem. Soc., 52, 3905 (1930).
76
T H E P R O P E R T I E S OF G A S E S AT H I G H P R E S S U R E S W H I C H CAN BE D E D U C E D F R O M E X P L O S I O N E X P E R I M E N T S ~ By H. JONES INTRODUCTION
Following the successful theory of the detonation of gases ~ontained in tubes formulated by D. Chapman (1) and E. Jouguet (2), at the beginning of the century, attempts have been made, at first notably by Schmidt (3), to apply this theory to solid explosives. In order to determine the velocity of the detonation wave theoretically a knowledge of the equation of state of the product gases of the explosion is required in a range of temperature and pressure far beyond those realizable in ordinary laboratory work. The pressures involved are of the order l0 s atmospheres and temperatures of the order of 3000~ Since the detonation velocity is a function of the loading density of the explosive, and since these velocities can be measured with considerable accuracy, it is natural that attempts should have been made to reverse the argument and to deduce information about the equation of state from the observed detonation rates. I t is the purpose of this paper to discuss the extent of the information concerning the equation of state which can be obtained in this way from detonation velocities. Care has to be taken when applying the ChapThe author is indebted to the Director-General of Scientific Research (Defence), Ministry of Supply, for permission to publish thls paper.
man-Jouguet theory to rods of solid explosives because this theory applies only to a strictly one dimensional detonation wave. It has been shown (4) that the lateral expansion in the reaction zone of a detonation wave causes a diminution in the velocity, which is given by an expression of the type
(Uo/U)' = 1 + f(l/r)
(1)
where f is a function which can be calculated approximately and which diminishes rapidly as l/r diminishes. U is the actual detonation velocity and U0 that which would be given by a correct application of the Chapman-Jouguet theory, r is the radius of the charge and l the length of the reaction zone. Thus the only data which should be used in an attempt to obtain information about the equation of state are velocities determined for charges of diameter large enough to give the maximum detonation rate. For most high explosives, charges of diameter of about one inch usually give the maximum velocity, but for the low power explosives, e.g. those containing much ammonium nitrate, the observed velocity will not at all correspond to the limiting value, which alone can be used with the Chapman-Jouguet theory to yield information on the equation of state.
B U R N I N G A N D D E T O N A T I O N OF E X P L O S I V E S
and
w THE PRESSURE AND DENSITY IN THE DETONATION f R O N T
OF l a G _ OF / O G
Let p denote the pressure and v the specific volume of the gaseous products in the detonation front, and let E(p, v) be the internal energy of the gas per unit mass measured relative to the energy of the unexploded material at atmospheric pressure, and Q be the chemical energy liberated per unit-mass by the reaction which, in general, is also a function of p and v. The well known equations of the Chapman-Jouguet theory can thus be written
E(p, v) - O = 89 p =
V2p(1
-
5Q1
- v)
(2)
pb),
(3)
where U is the detonation velocity and p the loading density of the explosive. The values of p and v in the detonation front are given, according to the Chapman-Jouguct theory, by the point in the p, v plane where, for given U and p, the straight line (3) touches the curve (2). Instead of following the usual procedure of assuming a form for the equation of state and thereby deducing p and v and hence U for each value of p and comparing this calculated dependence of U on p with experiment, we regard U as a known function of p and from this obtain directly information about the equation of state. Regarding p, v and p as independent variables equation 3 determines a ruled surface which touches the curved surface (2) along a certain line. According to the Chapman-Jouguet theory the values of p and v along the line of contact determines the pressure and specific volume in the detonation front for any given value of the loading density p. The condition that the surfaces (2) and (3) touch leads to two equations which express the fact that at any point of contact the direction cosines of the normal to each surface are equal. It is convenient to write 1/p = z, and g(z) = U2p2 which is to be regarded as a function of z, and then to express equations 2 and 3 in the form
F(v, p, z) =-E (p, v) -- O(p, v) -- 89p(z -- v) = 0
(4)
G(v, pm z) = g(z)(z - v) - p = 0
(5)
bz / oz
(7)
Ov /
Equation 6, together with equations 4 and 5, gives
0 Or ( E -
Q) + 8 9
pp ( 0 1 -pv.~(E-Q)
which together with equation 4 determines p and v when E and O are known. Equation 3 then determines the detonation velocity for any given p. This leads, therefore, to the usual ChapmanJouguet theory. Equation 7, on the other hand, leads to a result which does not appear to be well known. From equations 4, 5, and 7, we find
89
T,,
9(1 +
-
~ ]
(9)
Writing (6/~)v)(E - Q) = (1/~)p, which defines the quantity a, equation 9 together with equation 5 gives easily
U=p P =
{ (2+a)
l+d
d log U \ log p j
(10)
and
=
1(2 +
1+
dlogU (11) log p)_l
Thus apart from a, which we discuss below, these equations determine the pressure and the specific volume in the detonation front, from the observed U, p relation, without any assumption concerning the equation of state. Another result of the Chapman-Jouguet theory is that the fluid velocity, relative to the detonation wave, is just equal to the local velocity of sound. In symbols this relation is
The conditions for contact are then given by --V2(bp /Dv), = p2v~U~
OF /
OF _ 0(7 /
a;/oi,
OO
o,/op
(12)
(6) where (i~p/Sv), denotes the adiab~,tic variation of
592 pwithv.
T H I R D SYMPOSIUM ON COMBUSTION~ ]FLAME AND E X P L O S I O N P I t E N O M E N A
Thus if "r =
\ 0 log v/~
(13)
so that y is the initial value of the exponent in the adiabatic relation/~'2 = const., then we have from equations 10, 11 and 12
= (2 +
1+
d log U l
ag-og, J -
1.
(14)
We now show that a is essentially positive and therefore equation 10 gives an upper limit to p with a = 0, and similarly equations 11 and 14 give lower limits to v and ~,. The quantity E - O is the internal energy of the gaseous products relative to a standard state in which the constituents are taken as elements in their normal state. Thus since
it is clear that c~ is positive, for both the energy and the volume are increasing functions of T at constant pressure. The specific heat at constant pressure for the gas mixture is given by
Hence (15) and (16) together with the equation defining a give a={-~(~-v)
- - 1 } -1.
(17)
Equation 13 together with the well knov~n thermodynamic relation for the ratio of the specific heats gives C,(0T)
=
7C,,(Oj)
(18)
x~.here C~ is the specific heat at constant volume. Thus we can put finally
a = [y,(Op/OT).
1
(19)
a is, of course, a function of p and v and therefore equations 10 and 11 do not give p and v explicitly; the usefulness of this form is due to the fact that 0 < a < < 2 and therefore even without any
knowledge of the equation of state we obtain approximate formulae for p and v. Using equation 19 the value of ~ can be obtained approximately, since the specific heat at constant volume can be rather accurately estimated from the well known specific heats of the product gases at high temperatures. An uncertainty arises on account of the high pressure. The effect of pressure, however, can only increase Cv, hence omitting the effect of high pressure will tend to over estimate a. To estimate v(i)p/bT)~ the equation of state deduced by Becker (5) for nitrogen was used. From these estimates it becomes clear that the least value which is at all likely for .rC~,/(Opv/~)T)~ is about 5 so that = is not more than 0.25. Thus it can be seen that, if the observed relation between U and p is used, it is possible to find rather accurate and reliable values of the pressure and density in the detonation wave front as well as the initial value of the exponent for adiabatic expansion, in spite of the lack of information about the equation of state in the range of p and v which exists in the detonation wave. With regard to the temperature of the detonation front, it is clear that the U, o relation cannot determine it since only the variables p and v occur in the fundamental equations. The temperature can, however, also be calculated with fair accuracy and reliability because it is largely determined by the specific heats of the product gases, which for low pressures are well known, and by the heat of the reaction Q. However, one cannot determine theoretically the composition of the products without knowing the equation of state, so that an uncertainty arises in this respect. Also one cannot infer the composition of the products from closed bomb experiments, since the products after adiabatic expansion will, in general, be different from those in the detonation front since chemical reactions will proceed during the expansion. However, as a rule fairly reliable values of T can be obtained which are not very sensitive to the assumed form of the equation of state. I t is now possible to appreciate the limited information which can be deduced from an observed U, o relation. We can expect to obtain a set of p, v, T values, for the products of detonation, over a certain limited range of a single variable. The equation of state of the gas mixture may be expressed as
f(p,
v, 2", N a , 2VB, - . . ) = 0
(20)
where the N a etc. are the mole fractions of the gas
BURNING
AND DETONATION
mixture. When the NA" 9 9 are held constant the equation 20 represents a surface in the p, v, T space. The maximum amount of information which can be derived from a U, p relation would fix a line of a certain length in this surface. Thus if one attempted to analyse the detonation velocity data by first assuming an equation of state, and allowed this equation to contain a large number of undetermined parameters which could, in principle, be evaluated from the U, p relation, yet the resulting equation of state would be very largely arbitrary since there are evidently any number of surfaces which contain the same line. The conclusion is that very little information about the form of the equation of state can be obtained from detonation velocity data but that if the form is assumed then the values of parameters in the equation can, in general, be determined. w
593
OF EXPLOSIVES
" 4, ~. ~, ~. &2
o
,
z
~
r
.
i ~,,o* i$
Fro. 1. Curve (a) determined by the detonation velocity-loadlng density relation. Curve (b), BridgII2an~s measurements on nitrogen.
f'*
1.2
N U M E R I C A L R E S U L T S I N T H E CASE OF P E N T H R I T E
As an illustration of the kind of information which can be obtained with the assumption of a particular form of the equation of state, this section gives the results of a calculation based on the U, p data published by Friedrich (6) for the explosive penthrite CsHsOI2N4. The products of detonation may be assumed to consist of CO~, CO, H~O, H2, N2; the relative amounts being determined by the assumption of thermal equilibrium3 Whatever the relative amounts of these gases may be, one gm. moh of penthrite will always yield 11 moles of the gas mixture, A very simple form for the equation of state of the gas mixture was chosen, viz.
pv = R T + f(p)
(21)
w h e r e f is some unspecified function of the pressure alone which is to be determined from the U, p relation, R is the gas constant per gin. of the gas mixture. The total energy of the gas, allowing for the variation of the composition, was calculated as a function of T u s i n g specific heats determined from spectroscopic data, and was found to be very nearly linear with T in the range 2000 ~ to 4000~ When the equation of state is of the type (19) and the energy is linear in T, ~ becomes a constant, . and in the present case was found to have a value
,?, ,.0 O
~ ae
> c~ ~o 0.4-
a2
to
20
5o
&o
5o
Fro. 2. Potential energy of molecular repulsive forces. 0.200. Hence the pressure in the detonation front, as a function of loading density, is obtained immediately from equation 10 and a simple calculation shows that f(p) is given by the following formulae: Ot
f(p) = r
_(a/l+a)
+ ~,p op p-(1+2,~/l+,,)g, dp
2 Calculations of this type are given in detail in a forthcoming paper to appear in the Proc. Roy. Soc. by H. Jones and A. R. Miller.
6o
CUBIC A.U. PER MOLECULE
(22)
where ~b is a function of p given in terms of the
594
T H I R D SYMPOSIUM ON COMBUSTION~ F L A M E AND E X P L O S I O N P H E N O M E N A
parameter p by equation 10, and (1-4-a)r
v2 2(2+a)
( 1-t-2d
dl~
9
1
-4- 6 1 ~ , /
dlog U~ logo,/ (23)
where Q is the constant heat of reaction. Using the experimental data given by Friederich (6) (1933) the values of (pv - RT) have been calcula[ed as a function of p for pressures up to 280,000 atms. Figure 1 shows the results together with Bridgman's (7) observed values for N= at 68~ up to 15,000 atm. An equation of state of the type (21) leads to a term in the internal energy of the form
which is the energy arising from the repulsive forces between the molecules. Figure 2 shows this energy as a function of the molecular volume.
If the equation of state, thus obtained, is now used to determine the adiabatic law and hence to obtain the initial shock wave velocity from the end of a bare charge, very good agreement is obtained with experimental values. This, however, is again no justification for the assumed form of the equation of state because the iDitial shock wave velocity is determined mainly by just those parts of the p, v, T surface which are determined by the observed U, p relation. REFERENCES
1. CHAPMAN,D. L.: Phil. Mag, 47, 90 (1899). 2. JOUOUET,E.: J. Maths. pures appl., 6 (Series II), 5 (1906). 3. Scrr~io'r, A.: Zeit. ges. Schiess-Springstoffw.,31, 8, 37, 80, 114 (1936). 4. JONES, H.: Proc. Roy. Soc., A189, 415 (1946). 5. BEC~ER, R.: Zeit. fur Physik, 4, 393 (1921). 6. FRIEDERmH, W.: Zeit. ges. Schiess-Sprengstoffr 28 (1933). 7. B~DG~N, P. W.: Proc. Amer. Acad., 49 (1913); 9 (1923); 59 (1923).