Hugoniot’s equation

H) W. Fickett W.W. Wood, The Physics of Fluids 1 (6), 528-34 (Nov-Dec 1958) (Detonation-product equations of state, known as "constant-/ and "constant-)/ , obtained from hydrodynamic data) I) J.J. Erpenbeck D.G. Miller, IEC 51, 329-31 (March 1959) (Semiempirical vapor pressure relation based on Dieterici s equation of state J) K.A. Kobe P.S. Murti, IEC 51, 332 (March 1959) (Ideal critical volumes for generalized correlations) (Application to the Macleod equation of state) Kj) S. Katz et al, jApplPhys 10, 568-76(April 1959) (Hugoniot equation of state of aluminum and steel) K2) S.J. Jacobs, jAmRocketSoc 30, 151(1960) (Review of semi-empirical equations of state)... 

These equations can be combined to eliminate the velocities, yielding the Rankine Hugoniot equation for internal energy jump in terms of pressures and specific volumes (V s 1/p)... 

Accdg to Dunkle s Lecture delivered at Picatinny Arsenal on Dec.13, 1955, Hydro-dynamic Theory of Detonation , (Ref 78), utilizes the laws of conservation of mass, energy and momentum to derive certain relationship known as the "Rankine-Hugoniot Equation . There are five basic equations, of which. the first three are related to five variables pressure, specific volume, energy, detonation velocity and particle velocity... 

CA 44, 1032(1950) (Penetrating or piercing jet theory of deton) 39) S.R. Brinkley Jr, "The Theory of Detonation Process , pp 83-8 in the "Summary Technical Report Division 8, NDRC , Vol 1 (1946). It includes Riemann formulation (pp 83-4 86) Rankine-Hugoniot condition ( 84 86) Chapman-Jouguet postulate (84) Becker semiempirical equation of state (85) Rayleigh, solution of the Riemann equation (86) and Hydro-thermodynamic theory, applications (87-8) 40) W. Loring H. 

Dunkle s Lecture at Picatinny Arsenal, 21 Nov 1955, p 3 (One-dimensional steady-state process and Rankine-Hugoniot equation) 78) Ibid, 13 Dec 1955, p 5 (Nature of shock waves) p 8 (Chapman-Jouguet point) pp 8-9 (Basic equations of deton) p 9... 

Dunkle s Syllabus (1957 58), 1-36 (Detonation phenomena, mathematical background) 37-60 (Initiation of shock waves formulas equations including Riemann equation, p 43 Hugoniot relations in gases, p 44 Rankine-Hugoniot equation, p 45 ... 

As discussed in Vol 7, HI 79, the Hugoniot equations (or more correctly Rankine-Hugoniots), the simultaneous measurement of any two of the above variables is sufficient to determine all the rest provided conditions ahead of the shock (u0s P0, p o> Eo To) are known. Thus, for mathematical convenience but closely approximated in reality, the shock abruptly divides virginal (unshocked) material from shocked (compressed) material. For the reader s convenience, because we will refer to them frequently, this is illustrated in two graphs taken from the above Vol 7 article. They show the transmission of a shock from one material to another (script S s are shocks and script R s are rarefactions)... 

Both 2m and s may be measured quite accurately by a variety of techniques such as precisely spaced pins that close electrical circuits and high-speed cameras. Then, from Eqs. (16) to (19) and the initial conditions, one can And P, E, and V for the compressed material behind the shock front and the equation of state E(P, V) of the material near the Hugoniot curve. Various other reasonable assumptions ultimately permit fairly accurate determinations of E(P, V) for pressures and densities further removed from the Hugoniot curve. For each value of P and V, a separate experiment producing particular values of x and m is needed. 

Fig. 7.16. Experimental and calculated equation of state for (a) MgO and (b) CaO taken from Bukowinski (1985). Both calculated isothermal compression data and Hugoniot data (with shading representing uncertainty due to possible errors in the Gruneisen parameter) are shown. See original text for details of sources of experimental data (after Bukowinski, 1985 reproduced with the publisher s permission).

Example 20.2 In Example 20.1, we saw a hypothetical explosive that had the CJ state properties po = 1.43 g/cm , D = 6.95 km/s, andPcj =19.15 GPa. What is the equation of the left-going shock wave P-u Hugoniot of its detonation reaction products ... 

Time of arrival is easily calculated because below 2-3 kb, the wave velocity closely approaches the standard acoustic velocity ( 1500 m/s). For close-in timing, velocities are found from the P-u and [f-u Hugoniots for water or a convenient equation of state such as that from Rice-Walsh (Ref. 28)... 

In the present approach, we apply an accurate and numerically efficient equation of state for the exp-6 fluid based on Zerah and Hansen s hypemetted-mean spherical approximation (HMSA) [111] equations and Monte Carlo calculations to detonation, shocks, and static compression. Thermal effects in the EOS are included through the dependence of the coefficient of thermal expansion on temperature, which can be directly compared to experiment. We find that we can replicate shock Hugoniot and isothermal compression data for a wide variety of solids with this simple form. 

Let X s x(t) be the equation of a line of discontinuity. Then across it the Eankine-Hugoniot jump conditions hold. 

Fig. 1. Van der Waals equation of state for water. The saturation (binodal) curve and the spinodal curve are shown as dashed lines. A metastable initial state in the fluid is indicated by point A, superheated with respect to the boiling point at that pressure, indicated as point A . A steady detonation shock produces state B on the Hugoniot curve subsequent relaxation to the initial pressure at point C takes place along the isentrope labeled S.

Table S10.15 Coefficients in Hugoniot equations for molecular compounds c in km/s [10.205]...

The HOM equation of state described in Appendix A was used to calculate the Hugoniots for partially reacted explosive. The parameters used for 9404 are given in Table 4.1 and on the CD-ROM. Ramsay s unreacted equation of state. Us = 0.2423 + 1.883f7p, was used to describe the unreacted 9404 explosive. The BKW equation of state described in Appendix E was used to describe the 9404 detonation products. 

Two sets of C and S coefficients are sometimes necessary to fit the experimental Hugoniot data. For Vs < MINV, the fit Us = Cl + 51 Up is used. For volumes greater than Vo the Grtineisen equation of state was used along with the P = 0 line as the standard curve. 

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