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An alternative method for calculating gurney velocities
COMBUSTION AND FLAME 34:213-214 (1979)
213
An Alternative Method for Calculating Gurney Velocities MORTIMER J. KAMLET and MILTON FINGER
Naval Surface Weapons Center, White Oak Laboratory, Silver Spring, Maryland 20910 and Lawrence Livermore Laboratory, Livermore, California 94550
Assuming that an explosive should deliver a characteristic specific energy, Ee, for driving metal, Gurney has derived a series of equations for estimating fragment velocities for a variety of metal/ explosive configurations [1]. For the example of an explosives-filled metal cylinder, the metal velocity, U is expressed as
average molecular weight of these gases, and Q is the heat of detonation in calories per gram. 1 Taking the form of (3) and a least-squares fit of detonation pressures computed for a variety of explosives by the TIGER code to arrive at a new value of K, Hardesty and Kennedy [2] have defined a characteristic velocity, c* (mm//asec), as
(4)
(1) 2N/~e =
+
,
where m and c are the mass per unit length of the metal and explosive. The ~ quantity, which has the units of mm//asec, is known as the Gurney velocity and is characteristic of a given explosive. Hardesty and Kennedy [2] have shown that Gurney velocities are reasonably well approximated by TIGER computer-code calculations of energy to three-fold expansion along the detonation isentrope: 2X/~e = [x/2(Eo - E s ) ] V/Vo=3.
(2)
They have also related the Gurney velocity to the characteristic ~0 quantity used by Kamlet and Jacobs [3] to calculate Chapman-Jouguet detonation pressures
They next carried out a least-squares correlation of c* with values of [ X / ~ o - E ~ ) ] ~=3 for a number of explosives at densities from 0.4 g/cc to crystal densities to obtain their equation for the Gurney velocity in terms of ~o: 2X~e=0.6+0.5~o
K=15.58,
~o=NMalZQ 1/2,
(3)
where N is the number of moles of gaseous detonation products per gram of explosive, M is the Copyright © 1979 by The Combustion Institute Published by ElsevierNorth Holland, Inc.
(5)
We have followed a somewhat different path to arrive at a method for calculating Gumey velocities. Since Pj at the top of the isentrope depends on the square of the initial density [equation (3)] and the volumetric heat of detonation after expansion to STP (the bottom of the isentrope) depends on the first power of the density, energy delivered at various stages of the isentropic expansion should be calculable by equations of the form
Evolumetric =K~oPon, Pj=K~opo 2,
•
1
(6)
1 These quantities are hand-calculated by the H20-CO z "arbitrary," where oxygen, as available, first burns hydrogen to water, and then carbon to carbon dioxide [4].
0010-2180/79/020213+2501.75
214
MORTIMER J. KAMLET and MILTON F I N G E R TABLE 1 Estimation of Gurney Velocities Equation 7:(2Eg)0.5 = 0.887~oo.5p0o.4 Equation 5:(2Ee)0.5 = 0.60 + 0.54(1.44~o00)o.5 Gurney Velocity (mm//asec)
Explosive
PO, g/cc
¢
Measureda
Eq. (7)
Eq. (5)
TNT Nitromethane HMX RDX Tetryl PETN RDX/TNT-64/36 RDX/TNT-77/23 HMX/TNT-78/22
1,63 1.14 1.89 1.77 1.62 1.76 1.717 1.754 1.821
4.838 6.555 6.772 6.784 5.615 6.805 6.063 6.319 6.325
2.37 (2.44) 2.41 2.97 2.93 (2.83) 2.50 2.93 2.71 2.79 2.83
2.37 2.39 2.98 2.90 2.55 2.90 2.71 2.79 2.84
2.42 2.37 2.92 2.85 2.55 2.84 2.69 2.75 2.80
TIGER Code
{[J2(Eo = E~)] V / V o
=
31
2.40 2.42 2.98 2.90 2.66 2.85
a See Hardesty and Kennedy [2] for sources of these experimental results.
Fitting STRETCH.BKW computer-code results [5] and Lawrence Livermore Laboratory (LLL) cylinder test measurements to (6), we have found the best correlations o f energies delivered between --- 2 and V = 7 to be with n = 1.8. 2 Converting the resulting equation to a gravimetric basis, taking the square root, and fitting to LLL cylinder test results for TNT to estimate a value for K, we have arrived at the somewhat simpler equation for Gurney velocities: V~e
= 0,887~0°'5Po°'4
(7)
Measured Gurney velocities [from cylinder test results and equation (1)] are compared in Table 1 with calculations b y the present method, the HardestyoKennedy method, and the TIGER code. It is seen that the present method approximates the measured results most closely [average difference = 0.02 mm//~sec for equation (7), 2 We will discuss the cylinder test correlations in greater detail in a future paper to be submitted to this journal,
0.04 mm/psec for (5), and 0.05 mm/#sec for the TIGER calculations].
Work by M. J. K. was done under Naval Surface Weapons Center Independent Research Task 1R-144. We wish to acknowledge the kind assistance o f Dr. J. Kennedy o f the Sandia Laboratories.
REFERENCES 1. 2. 3. 4. 5.
Gurney, R. W., The Initial Velocities of Fragments from Bombs, Shells, and Grenades, BRL Report No. 405 (1943). Hardesty, D. R. and Kennedy, J. E., Combust. Flame 28, 45 (1977). Kamlet, M. J. and Jacobs, S. J., J. Chem. Phys. 48, 23 (1968). Kamlet, M. J. and Ablard, J. E., J. Chem. Phys. 48, 36 (1968). Mader, C. L., Detonation Properties of Condensed Explosives Computed Using the Becker-Kistiakowsky-Wilson Equation of State. Los Alamos Scientific Laboratories Report LA-2900 (1963).
Received 19 May 1978
213
An Alternative Method for Calculating Gurney Velocities MORTIMER J. KAMLET and MILTON FINGER
Naval Surface Weapons Center, White Oak Laboratory, Silver Spring, Maryland 20910 and Lawrence Livermore Laboratory, Livermore, California 94550
Assuming that an explosive should deliver a characteristic specific energy, Ee, for driving metal, Gurney has derived a series of equations for estimating fragment velocities for a variety of metal/ explosive configurations [1]. For the example of an explosives-filled metal cylinder, the metal velocity, U is expressed as
average molecular weight of these gases, and Q is the heat of detonation in calories per gram. 1 Taking the form of (3) and a least-squares fit of detonation pressures computed for a variety of explosives by the TIGER code to arrive at a new value of K, Hardesty and Kennedy [2] have defined a characteristic velocity, c* (mm//asec), as
(4)
(1) 2N/~e =
+
,
where m and c are the mass per unit length of the metal and explosive. The ~ quantity, which has the units of mm//asec, is known as the Gurney velocity and is characteristic of a given explosive. Hardesty and Kennedy [2] have shown that Gurney velocities are reasonably well approximated by TIGER computer-code calculations of energy to three-fold expansion along the detonation isentrope: 2X/~e = [x/2(Eo - E s ) ] V/Vo=3.
(2)
They have also related the Gurney velocity to the characteristic ~0 quantity used by Kamlet and Jacobs [3] to calculate Chapman-Jouguet detonation pressures
They next carried out a least-squares correlation of c* with values of [ X / ~ o - E ~ ) ] ~=3 for a number of explosives at densities from 0.4 g/cc to crystal densities to obtain their equation for the Gurney velocity in terms of ~o: 2X~e=0.6+0.5~o
K=15.58,
~o=NMalZQ 1/2,
(3)
where N is the number of moles of gaseous detonation products per gram of explosive, M is the Copyright © 1979 by The Combustion Institute Published by ElsevierNorth Holland, Inc.
(5)
We have followed a somewhat different path to arrive at a method for calculating Gumey velocities. Since Pj at the top of the isentrope depends on the square of the initial density [equation (3)] and the volumetric heat of detonation after expansion to STP (the bottom of the isentrope) depends on the first power of the density, energy delivered at various stages of the isentropic expansion should be calculable by equations of the form
Evolumetric =K~oPon, Pj=K~opo 2,
•
1
(6)
1 These quantities are hand-calculated by the H20-CO z "arbitrary," where oxygen, as available, first burns hydrogen to water, and then carbon to carbon dioxide [4].
0010-2180/79/020213+2501.75
214
MORTIMER J. KAMLET and MILTON F I N G E R TABLE 1 Estimation of Gurney Velocities Equation 7:(2Eg)0.5 = 0.887~oo.5p0o.4 Equation 5:(2Ee)0.5 = 0.60 + 0.54(1.44~o00)o.5 Gurney Velocity (mm//asec)
Explosive
PO, g/cc
¢
Measureda
Eq. (7)
Eq. (5)
TNT Nitromethane HMX RDX Tetryl PETN RDX/TNT-64/36 RDX/TNT-77/23 HMX/TNT-78/22
1,63 1.14 1.89 1.77 1.62 1.76 1.717 1.754 1.821
4.838 6.555 6.772 6.784 5.615 6.805 6.063 6.319 6.325
2.37 (2.44) 2.41 2.97 2.93 (2.83) 2.50 2.93 2.71 2.79 2.83
2.37 2.39 2.98 2.90 2.55 2.90 2.71 2.79 2.84
2.42 2.37 2.92 2.85 2.55 2.84 2.69 2.75 2.80
TIGER Code
{[J2(Eo = E~)] V / V o
=
31
2.40 2.42 2.98 2.90 2.66 2.85
a See Hardesty and Kennedy [2] for sources of these experimental results.
Fitting STRETCH.BKW computer-code results [5] and Lawrence Livermore Laboratory (LLL) cylinder test measurements to (6), we have found the best correlations o f energies delivered between --- 2 and V = 7 to be with n = 1.8. 2 Converting the resulting equation to a gravimetric basis, taking the square root, and fitting to LLL cylinder test results for TNT to estimate a value for K, we have arrived at the somewhat simpler equation for Gurney velocities: V~e
= 0,887~0°'5Po°'4
(7)
Measured Gurney velocities [from cylinder test results and equation (1)] are compared in Table 1 with calculations b y the present method, the HardestyoKennedy method, and the TIGER code. It is seen that the present method approximates the measured results most closely [average difference = 0.02 mm//~sec for equation (7), 2 We will discuss the cylinder test correlations in greater detail in a future paper to be submitted to this journal,
0.04 mm/psec for (5), and 0.05 mm/#sec for the TIGER calculations].
Work by M. J. K. was done under Naval Surface Weapons Center Independent Research Task 1R-144. We wish to acknowledge the kind assistance o f Dr. J. Kennedy o f the Sandia Laboratories.
REFERENCES 1. 2. 3. 4. 5.
Gurney, R. W., The Initial Velocities of Fragments from Bombs, Shells, and Grenades, BRL Report No. 405 (1943). Hardesty, D. R. and Kennedy, J. E., Combust. Flame 28, 45 (1977). Kamlet, M. J. and Jacobs, S. J., J. Chem. Phys. 48, 23 (1968). Kamlet, M. J. and Ablard, J. E., J. Chem. Phys. 48, 36 (1968). Mader, C. L., Detonation Properties of Condensed Explosives Computed Using the Becker-Kistiakowsky-Wilson Equation of State. Los Alamos Scientific Laboratories Report LA-2900 (1963).
Received 19 May 1978