Multi-phase Explosion Detonation and JWL EOS Parameters Numerical
Calculation
He Ning1,a , Zhang Qi2,b
1
2
North China Univ. of Science and Technology, Yanjiao 101601, Beijing, China
Beijing Institute of Technology, National Key Laboratory of Explosion Science and Technology, Beijing,
100081, China
a
ning_95@yahoo.com.cn, bqizhang@bit.edu.cn
Keywords: Multi-phase Explosion, JWL EOS, Detonation parameter, Numerical simulation
Abstract. According to the chemical balance and the minimum free energy principle calculate
explosive products and Detonation parameters, and then BKW equation was utilized to calculate JWL
EOS parameters. Visualization software of Detonation and JWL EOS parameters was development,
which was provided with favorable user interface. The calculation results and experimental data were
basically consistent. The paper solved the bottleneck problems regarding the technology of numerical
simulation of dynamic response and damage effect of Multi-phase explosion of explosive hazard
sources.
Introduction
With the development of computer technology, the calculation could be programmed and the results
are given quick and easy. As it is complicated to define the constitutive equation, it is impossible to
numerically simulate the process of damage effect of explosion for most substances of explosive
hazard. In existing calculation of dynamic response of explosion, data regarding the EOS parameters
of a few typical explosives are available. For explosive substances with substantial differences in
characteristics of explosion energy, there will be great error if the method of TNT equivalent is used
to analyze response of explosion in the near-by area of the explosion. As a result, it is the pressing
fundamental problem in the fields of explosion science to obtain the constitutive equation of general
explosive substance hazard. Some scientists have tried, but no mature method is found. It is still the
basic problem in the front of explosion science to obtain constitutive equation of explosion from the
components of explosive substances directly.
Numerical Calculation of Detonation and JWL EOS Parameters
Detonation is a limit state of explosion. According to the chemical balance and the minimum free
energy principle calculate explosive products and Detonation parameters.
 p 
c2   2 
  2  S , Considering the C-J point, which can be derived:
Sonic after wave front is
u1 
 p2 
 2 c2 1



1
1
 1 /  2  S

u1  2  2 R2T2
1
According to isentropic gas relation is:
(1)
(2)
This is a very useful formula, but because of the unknown parameters of the region already, it still
cannot determine the Temperature. The physical state of conservation of energy of the fired zone is


1
 p2  p1  1  1   q  H 2  H1
2
 1  2 
determined by the Hugoniot equation:
(3)
JWL equation of state (EOS) was advanced by Lee E.L. with Lawrence Livermore National
Laboratory in 1965 based on the work of Jones and Wilkins[1].“Cylinder test” is generally required to
solve the six parameters of Detonation and JWL EOS[2]. Fluid dynamic programme can also be used
to calculate the process of cylinder test through numerical simulation[3]. For expensive or
newly-developed explosive, such definition of parameters is more difficult. Therefore, it is the
fundamental problem confronting the field of explosion science to obtain constitutive equation of
explosion directly from the components of explosive substances. Both the field of explosion science
and the field of public security are in urgent need of a more convenient and effective numerical
calculation method to calculate JWL EOS parameter of explosives.
JWL EOS is as follows:


P  A1 
 R1V
  R1V


e
 B1 

 R2V
  R2V E
e

V

(4)
Where, P refers to the pressure of detonation products; V refers to relative specific volume of the
detonation products; E refers to specific thermodynamic energy; subscript s refers to the isentropic
process; A , B , C , R1 , R2 , and  refer to the six JWL EOS parameters to be defined[4].
D
P
E
Given explosion velocity J , explosion pressure J , and chemical energy 0 , of the given
explosive, the relation between the JWL EOS parameters can be obtained according to the CJ
condition and the conservation relation of mass, momentum, and energy. According to CJ
 P 
  S   0 D 2
V VJ
condition 
, it can be obtained that:
AR1e R1VJ  BR2e R2VJ  C1   VJ
Where,
VJ   /   1
and
   0 D / PJ  1
2
J
 1 
 0 D 2
(5)
, it can be obtained from the Hugoniot relation that:
A  R1VJ B  R2VJ C 
1
e

e
 VJ  E0  PJ 1  VJ 
R1
R2

2
(6)
CJ isentropic line passes through CJ point. Therefore,
Ae R1VJ  Be  R2VJ  CVJ
 1 
 pJ
(7)
The specific procedures are as follows:
First, define physical parameters of the explosive, such as composition, heat of formation, and
density.
Second, assuming a pressure P and a temperature T; then, P and T based on assumed equilibrium
calculation of the component (wave-front surface of the composition of detonation products); then,
according to energy conservation, the calculation of the energy balance of heat and pressure, if not
satisfied, return to “Assuming a pressure P and a temperature T “again to find suitable pressure P and
temperature T; If meet the equilibrium conditions, the results obtained; then obtained by calculating
the P, T calculated detonation velocity and other detonation parameters.
Last, substitute CJ detonation parameter calculated by BKW EOS into JWL equation and its
relation equations (4), (5), and (6), and iterate repeatedly till differences between the left and right of
the four equations are less than a given value. If there are negative values in iteration, they shall be
  Aig  Ai j
corrected by the damping factor  , so as to ensure smooth proceeding of iteration. If i
,
g
g
g
j
Z i  Ai   i  Ai   ( Ai  Ai )
, and Z i will be regarded as the initial value of next iteration.
Development of Visualization Software of Detonation and JWL EOS Parameters
Visualization software was furnished with direct and concise interface. Any one who is familiar with
simple operation of computer may learn the use of the software as quickly as learning the simple
office software such as note. The software employed graphic user interface for input, and required
parameters have been provided with default value. User may make corresponding modifications to
perform new calculation. The specific calculation process is as follows:
(1) Input initial environmental parameters: environmental parameters include initial temperature
and pressure, whose default values are 298 K and 1 atm.
(2) Input parameters of the explosive: input formation heat and density of the explosive, whose
default values are 73220J/mol and 1.64g/cm3. In the graphic user interface, the explosive may
comprise multiple elements.
Taking TNT for example as shown in Fig. 1 and Fig. 2 , 1 mol TNT contains 7 mol C, 5 mol H, 6
mol O, and 3 mol N, which were parameters to be input. Content of other elements was 0 mol.
Fig.1 Software Interface
Fig.2 Software results
Discussion
JWL EOS parameters of TNT explosive are obtained utilizing the numerical calculation method
provided in this paper. The calculation results basically agree with the experimental results after the
comparison between the calculation results and the experimental results provided in [5] (see Table 1
and Table 2).
Table 1 The JWL EOS parameters of TNT explosive
Explosive
TNT1[7]
A/GPa
3.712
B/GPa
0.03231
C/GPa
4.15
R1
0.95
R2
0.3
TNT2
3.85
0.04
4.2
1.21
0.3
Note: TNT1 includes the data provided in [6], while TNT2 includes the data calculated by this software.
Table 2 The relative error of JWL EOS parameters of TNT explosive (unit:% )
Explosive
A
B
R1
R2

TNT
3.4
23.8
1.2
27.3
0
Two groups of JWL EOS parameters of TNT explosives are embedded in DYAN for numerical
simulation and the calculation model is shown in Fig. 3. Fig.4 shows the dimensional schematic
diagram of the model, and Fig. 5 and Fig. 6 show the pressure and displacement of the same point on
steel plate.
Fig.3 Calculation model Fig.4 Schematic diagram of model
Fig.5 Pressure of the same point on steel plate Fig.6 Displacement of the same point on steel plate
Conclusions
This paper numerically calculates Detonation and JWL EOS parameters of Multi-phase explosive and
develops visualization of detonation parameters and Detonation and JWL EOS parameters. Verified
by examples, the numerical calculation results basically agree with the experimental results, and the
error range meets the actual engineering requirements.
Numerical calculation of Detonation and JWL EOS parameters can apply to explosives containing
11 elements, namely, C, H, O, N, Cl, F, S, Na, K, Al, and Mg. It solves of the problem regarding
calculation of detonation parameters of explosive containing only C, H, O, and N with the help of
preceding programs and lays foundation for research and application of explosives containing
multiple elements. The Detonation and JWL EOS parameters of explosives, reduces experimental
consumption to define parameters, and solves the bottleneck problems regarding simulation
technology of dynamic response and destruction value of explosion of general hazardous explosive
sources.
Acknowledgement
It is a project supported by the National Natural Science Foundation（107720229）.
References
[1] Jacobs S J. On the equation of state for detonation products at high desity.12th Symposium
(International) on Combustion, Pittsburgh. The Combustion Institute, 1969.
[2] Chen Lang, Long Xinping, Feng Changgen, and Jiang Xiaohua. Detonation of Aluminized
Explosive [M]. National Defense Industry Press, 2004, 6:42.
[3] Sun Chengwei, WeiYuzhang, and Zhou Zhikui. Applied Detonation Physics [M]. National
Defense Industry Press, 2000, 12:292-293.
[4] Kury J W,et al.Metal Acceleration by Chemical Explosives,4th Symp. on Detonation, 1965:3-13.
[5] Zhang Guanren and Chen Danian. Initiation Dynamics of Condensed Explosive [M]. National
Defense Industry Press, 1991, 9:134.