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EQUATIONS OF STATE FOR DETONATION PROBLEMS

William C. Davis CONI=- 9 7 0 7 0 7 - - / /

Los Alamos National Laboratory Los Alamos, NM 87545

The words “Equation of State” have usually been used in the detonation physics community to mean whatever information about the materials involved that would make it possible to solve the problem at hand. As the problems have become more complicated and the computer simulation more detailed, better and better equations of state have been developed and used. Still, most equations of state have been limited to use in solving the hydrodynamic equations for an inviscid, non-heat-conducting, non- rahating, non-reacting fluid. For this purpose, an incomplete equation of state of the form p = p@,v), with pressure given as a function of energy and specific volume, is adequate. It allows one to determine the three important derivatives, the adiabatic gamma y, the Griineisen gamma r, and the fundamental derivative 9, whch determine the properties of the fluid. All the information in such an equation of state is about the mechanical properties. There is no information about the temperature, entropy, or specific heat. It is adequate for calculating the performance of devices driven by ideal explosive.

A complete equation of state is one that employs the complete potential, and the potential best suited for hydrodynamics is E = E(S,v), the specific internal energy as a function of entropy and specific volume. With this equation of state the entropy, temperature, and specific heat can be obtained. With the present emphasis on calculating reaction zones of non-ideal explosives and insensitive explosives, DDT, critical diameter, partial reaction, and so forth, the complete equation of state is needed to provide temperature for the reaction rate. New experimental techniques, such as neutron scattering, will provide experimental values for the temperature to use for calibrating the equations of state.

thermodynamically consistent. That is, the incomplete equation of state determines the differential T dS = dE + p dv, so that the entropy can be obtained if an integrating factor UT can be found for the right hand side. The integrating factor is not unique, so there may be more than one complete equation of state corresponding to the incomplete equation of state. However, the complete equation of state must be thermodynamically consistent. That is, it must conform to all the thermodynamic constraints. It may happen that no choice of integrating factor yields such an equation of state. While many of the incomplete equations of state have been used quite successfully even though they are not consistent, they can not be actual physical descriptions of a real material unless they really correspond to an incomplete equation of state derived from a complete equation of state. The more demanding and detailed the problem, the more important it is to have a physical equation of state.

Many of the equations of state that have been widely used have not been

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Formulating a complete equation of state for a complicated material is not a simple problem. Fortunately it need not describe the material throughout all its possible states and phases, only those sampled by the flow in question. It must, however, be made perfectly clear where it works and where it doesn’t, and it must work over a large enough region to be useful.

In a paper in the Tenth Detonation Symposium [ 11 I presented a method for obtaining and calibrating an equation of state for hydrodynamic computations, that was a simple one of the form p = p(E,v). When I tried to extend it to include some of the thermal properties it was clearly unsatisfactory. At very high pressures, the specific heat tended to idinity. In the effort to find a better equation of state for problems where the thermal properties are important, several approaches have been studied. In this paper the constraints for a complete and consistent equation of state are reviewed. Some examples of equations of state are shown, and their derivation and physical basis explained. Some emphasis is put on how and why they fail to be satisfactory equations of state everywhere, and in what part of the thermodynamic space they can be used.

Menikoff and Plohr [2] have grven a very careful and complete discussion of the constraints on equations of state, and of the effects of features of the equation of state on the Riemann problem of the propagation of waves. Their results are the basis for the discussion in this paper of the constraints.

explosive, and another for the reaction products. A mixture rule is needed to combine the two. The conservation laws do not completely determine the state of the mixture. The total energy is given, but not how it is divided between the two materials. Additional assumptions are necessary to decide how to divide up the energy. Fortunately, in many cases the overall properties are relatively insensitive to the actual division, but sometimes it really matters. The mixture rules that have been employed in the past are discussed and critically reviewed.

The problems of calibrating complete equations of state have not been solved for those applied in the detonation region. Some suggestions are given about calibration, but there is no way around the need for good experimental measurements of the thermal properties, and these do not yet exist.

The focus of this paper is not on providing a new equation of state that will enable us to solve all our problems, but to discuss the restrictions on equations of state, and the effect of those restrictions on their properties. Later, when there are more data, it will be time to search for the best equations of state for all the various problems. Probably no single form, let alone a single calibration, can ever be applicable in all the regions of pressure, density, and temperature of interest.

Detonation problems often require the use of one equation of state for the unreacted

1. W. C. Davis, Equation of state for detonation products, Tenth International Detonation Symposium, 1993, pp. 369-376.

2. Ralph Menikoff and Bradley J. Plohr, The Riemann problem for fluid flow of real materials, Reviews of Modern Physics, 61,75-130 (1989).