numerical-modeling-of-explosives

7/18/2019 Numerical-Modeling-of-Explosives

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Comparison of BKW and JWL Equations of State

for Explosion Simulations

INTRODUCTIONThe capability to predict the consequences of explosion through computer simulations is of

great practical importance. The physical processes involved in a detonation phenomenon,

though, are extremely complex. Therefore, the common practice is to resort to empirical or

semi-empirical Equations of State (EOS) describing the detonation properties of explosives.

Becker-Kistiakowski-Wilson (BKW) and Jones-Wilkins-Lee (JWL) are among the most

widely used EOS. The purpose of this work is to conduct a comparative study between them.

A special emphasis was given to the description of the contaminants dispersion caused by the

explosion (in addition to the usual interest in describing blast waves). In particular, we study

the implications of these differences within the framework of our current computational

methodology using the open source CFD toolbox – OpenFOAM 1.

The paper is organized as follows: we start with a brief review of the above EOSs. Then we proceed to comparative study using 1D shock tube computations and 3D explosion in open air

above the ground. This is followed by discussion of the results and conclusions.

THE EQUATIONS OF STATE

Analytic analysis of JWL, an empirical EOS, and BKW, a semi-empirical EOS, is presented

below. Both EOSs are calibrated against the experimental shock Hugoniot 2,3. Therefore, are

in wide use to describe the detonation products soon after the chemical reaction took place

(high pressure range). The form of the BKW EOS is based upon a repulsive potential

approach applied to the virial equation of state 2. Thus, it should also provide a good

description of the detonation products in the intermediate pressure range. Another important

difference between the two is that BKW is a full EOS, while JWL is classified as a partial

EOS, which means that it has no explicit temperature representation. For the reasons

mentioned above, one may think that BKW EOS may be more suitable for computational

simulation of explosion, especially because of its physical basis. The choice of the appropriate

EOS, though, should be according to the specific problem under consideration.

Review of the Becker-Kistiakowsky-Wilson (BKW) EOS

BKW is an equation of state of the form2

(1)

1

where

(2) Θ

P is the Pressure, V is the molar gas volume, T is the temperature and R is the gas constant.

,, and are adjustable parameters. is the covolume of the mixture, based on the

stoichiometric decomposition of the gas products and calculated as follows:

(3)

and represent the mole fraction and the covolume of each product species , respectively.

As mentioned previously, BKW constants are calibrated against the experimental shock

Hugoniot. Although it has a physical basis, deriving an accurate equation of state for aspecific high explosive (HE) requires separate calibration procedure (a general set of



calibrated coefficient does not exist). As a practical compromise, several calibrated sets

(,, ,) are commonly in use, and the final fitting to the requested HE is achieved by the

calculated covolume.

As an example, a set of BKW EOS constants for RDX2 is given in Table 1, as well as the

covolume factor for C-4 (mostly RDX) in 1.65 g/cc density4

. The values are suitable for usein BKW equation of state with molar gas volume in units of [cc/mole].

Table 1: BKW EOS constants for 1.65 g/cc RDX4

Θ [K]

0.16 10.91 0.50 400 333.83

We note that the BKW EOS is calibrated in the high pressure regime. However, due to its

physical basis, it is expected to provide a good fit also in the mid pressure regime.

Furthermore, the low pressure regime described adequately by the ideal gas EOS, which is

asymptote of the BKW analytic term.

Review of the Jones-Wilkins-Lee (JWL) EOS

JWL EOS can be expressed 5,6 in the following Taylor expansion form (Mie-Grüneisen)

(4) Γ ⋯

Where P is the pressure and E is energy per unit volume. Subscript "s" denotes reference to

isentropic compression or expansion (used as the reference state), while "T" denotes reference

to an adiabatic process. / is the relative volume. Here Grüneisen coefficient is

constant and equal to . Substitute to explicit expressions of and we obtain

(5) 1

exp 1

exp

is the chemical energy present in explosive. , , , and are JWL constants. The

purpose of the first exponential term in (5) is to describe the high pressure regime. Due to the

rapid decrease of this term with compression ratio the second term becomes dominant for the

high-intermediate pressure range. The third term is actually the EOS of a perfect gas. It

becomes a leading term for the lower pressures.

The JWL is calibrated to match the detonation velocity and pressure in cylinder test

experiments (admissible up to limited range of expansion 7) and is forced to behave as an

ideal gas at low pressures. JWL EOS set of constants for 1.59 g/cc RDX is given 8 in Table 2.

Table 2: JWL EOS constants for 1.59 g/cc RDX

989.0848 11.11902 5.167 1.0458 0.396143

Analytic Comparison between BKW and JWL EOS



A simple way to examine the fundamental differences between the two equations of state can

be done by comparing them in a static state (without kinetic energy at all). The calculation

relates to RDX properties using its calibrated constants given in Table 1 and Table 2. RDX

was chosen since it does not exhibits "after burn" phenomena (in contrast to TNT for

example), which means that all the energy is released during the detonation process. Thus,

there are no delayed effects during the simulation.

The pressure values versus relative volume of three EOS (JWL, BKW and ideal gas) are

presented in Figure 1. Notice that the BKW covolume value we found in the literature is for

RDX whose density is 1.65 g/cc (slightly different from 1.59 g/cc for the JWL data). Value of

0.396 (the same as in JWL data) has been used to calculate the temperature from the

BKW energy expression2 for every state (compression ratio). Both calculations are in regard

to total energy of 5.29 MJ and total mass of 1 . The ideal gas representation can be

calculated either based on the last term of JWL equation or based upon the BKW equation by

omitting the correction, both give almost the same result (discrepancy of 3.8% due to

density differences).

Figure 1: RDX Pressure vs. Relative volume. left: high

pressure regime; right: mid pressure regime.

As expected, in the compression area (relative volume of less than 1) the computed pressures

of JWL and BKW EOS are the same (because both of the EOS's have been calibrated against

the Chapman Jouguet pressure). On contrast, in the expansion area it can be seen clearly that

the BKW curve declines slower than the JWL one, therefore there is a significant difference

in the pressure value between them (maximum difference of 55% at ≅ 1.5). At the relative

volume equal to 10, the BKW pressure is higher by about 30% than the JWL pressure, while

the latter being practically equal to that of the ideal gas. At relative volumes higher than 50,

the differences become negligible (less than 10%).

Shock Tube Simulations

An open source toolkit OpenFOAM 1 is commonly used for various CFD computations. It has been proved to be a reliable and, what is also very valuable, an extendible platform. In order

to be used for our purposes (explosion modeling) it has to be augmented by an additional EOS

appropriate for the detonation products description. Two such equations were implemented

within OpenFOAM – JWL and BKW. The solver used for our computations is

rhoCentralFoam – the only solver within the package suitable for high-speed flows. It has to

be noted that the numerical methods of the type implemented within this solver suffer from a

limitation on the strength of a shock wave it can describe without getting unstable. The "rule

of thumb" is that the jump in pressure should be smaller than 10. Therefore, instead of

simply assuming that the detonation products occupy initially the volume of the solid

explosive, we have to resort to "equivalent" initial conditions. The volume of sphere

containing detonation products is larger than that of the solid explosive, while their mass is

the same. Their internal energy is equal to the chemical energy of the explosive.



We present in this section some computational experiments with a two-material shock-tube

problem. Their purpose is to further examine the commonalities and differences between the

JWL and BKW EOSs. We assume that the "tube" is 10m long, the membrane is located at

distance of 2m from its right end (see Figure 2). The right part of the tube is filled with air at

atmospheric pressure and temperature of 300. The tube is occupied to the left of themembrane by detonation products whose density is 50 times smaller than that of the solid

explosive, while the specific internal energy is the same (~5.9 /). This corresponds to

the pressure jump of about 750780, depending upon the EOS.

Figure 2: Shock tube model

Our first observation is that the same density and specific internal energy correspond using

these two equations of state to pressures that differ by ~3.5% and practically the sametemperatures. Another observation is that for the compression ratio of 50 the JWL EOS

becomes practically identical to that of the perfect gas, since the exponential terms in Eqs. (5)

become negligible. Therefore, all the results reported in this section for the JWL EOS can be

regarded also as those for the ideal gas. It can also be seen from Figs. 3,4 that neither shock

speed nor its strength (pressure jump) depend significantly upon the equation of state at any

time. The high temperature near the wall (see Figure 4) is a manifestation of the so-called

"wall-heating" phenomenon – a well-known numerical artifact.

Figure 3: Pressure development: on the left – . (just before the shock approaches

the right wall), on the right – . (just after the shock reflection from the wall).

Figure 4: On the left: pressure . after the membrane

removal, on the right – temperature at . .

The interface between detonation products and the air at three different time instances isdepicted in Figure 5. The fluid flow equations solved here are inviscid, i.e. any physical



dissipation/mixing are not taken into account. The "smearing" of the interface here is the

result of numerical diffusion only. Note that at time t=0.65ms the shock wave is approaching

the boundary (wall), there is a slight difference (about 3cm) in the location of the detonation

products/air interface as predicted by JWL and BKW equations of state. This observation

implies that, within the framework of the current methodology, these two equations of state

produce very similar results regarding also the contaminants dispersion due to explosions inopen air (this will be verified in more detail in the next session). However, whether or not this

is the case for the confined explosions needs to be investigated.

Figure 5: Interface between detonations products and air at

different time for BKW and JWL equations of state.

3D Open Air Explosion Simulation

This section contains a report regarding explosion simulation using, as in the previous section,

OpenFOAM toolbox. Again, the explosive considered is RDX (charge of 1kg). The initial

conditions are as presented in Figure 6: a spherical region (containing the detonation

products) of radius ~20cm is located at 1m above the ground. The compression ratio

considered is again 50, i.e. the density is 1/50 of that of the solid explosive. The internal

energy enclosed within this region is equal to the chemical energy of 1kg charge of RDX.This corresponds to the initial pressure of about 78.5GPa (BKW EOS) and 74.5GPa (JWL

EOS) and to the pressure jump of ~750 accordingly. We conduct the computations as long as

there are regions where the flow is supersonic. This occurs up until time t~2.0ms.

Figure 6: Initial location of the detontation products (under high pressure and temperature).

The two EOSs, within the current methodology, produce practically the same results

regarding the pressure field in general and blast wave strength/speed in particular (Figure 7).

Also the contaminants dispersion picture is practically the same in the two cases (Figure 8).

a b



Figure 7: Pressure distribution. (a) t=0.5ms (the shock reflected from the ground). (b) t=2.0ms

Figure 8: Detonation products distribution. (a) t=0.5ms. (b) t=2.0ms

CONCLUSIONS

Our first conclusion is that there are rather significant differences between the two equations

of state for the "high-intermediate" range of compression ratios. Also, the JWL EOS becomes

practically identical to that of the ideal gas already for compression ratio of 5÷10, while the

BKW EOS still differs slightly from the two for compression ratios up until 50÷100. The

explanation to this is likely to be the following: JWL EOS was designed and calibrated to

reliably represent high pressure regimes, while BKW EOS – for both high and "intermediate"

pressure states.

The main practical conclusions relevant for our current methodology with OpenFOAM are:

For the compression ratios under consideration of 50 and above, the JWL EOS is

practically the same as that of the ideal gas.

The BKW EOS results are slightly different at these compression ratios from the ideal

gas. These differences, however, are negligible for the open space explosion

simulations. It remains to be seen whether or not they are of any significance for

simulating confined explosions since the reflected shock was almost the same.

Another beneficial study to conduct is to obtain the initial conditions for the

OpenFOAM simulations by means of another method/code. This code should be

capable of representing strong blast waves, and, therefore, can conduct simulationsstarting from the compression ratio of unity. Once this capability is made operational,

our intention is to conduct the comparative study again.

REFERENCES

1. OpenFOAM User Guide. at <www.openfoam.org>

2. Mader, C. L. Numerical Modeling of Explosives and Propellants. (CRC Press: 2007).

3. Suceska, M., Ang, H.-G. & Chan, H. Y. S. Study of the Effect of Covolumes in BKW

Equation of State on Detonation Properties of CHNO Explosives. Propellants, Explosives,

Pyrotechnics 35, 1–10 (2010).

4. Lewis, M. W. & Wilson, T. L. Response of a Water-Filled Spherical Vessel to an Internal

Explosion. (Los Alamos National Laboratory LA-13240-MS: 1997).

a b



5. A. Alia, M. S. High explosive simulation using milti-material formulations. Applied

Thermal Engineering 26, 1032–1042 (2006).

6. Souers, P. C. JWL Calculating. (Lawrence Livermore National Laboratory: 2005).

7. Kerley, G. I. & Christian-Frear, T. L. Prediction of Explosive Cylinder Tests Using

Equations of State from the P A Code. (Sandia National Laboratories: 1993).

8. Grys & Trzcinski Calculation of Combustion, Explosion and Detonation Characteristics ofEnergetic Materials. Central European Journal of Energetic Materials 7(2), 97–113

(2010).

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