Abel equation of state

Very little has been published covering such nonideal, but very realistic, situations. Two publications by Wiederman (1986a,b) treat nonideal gases. He uses a co-volume parameter, which is apparent in the Nobel-Abel equation of state of a nonideal gas, in order to quantify the influence on flragment velocity. The co-volume parameter is deflned as the difference between a gas s initial-stage speciflc volume and its associated perfect gas value. 

Abel Equation of State. See under Detonation (and Explosion), Equations of State... 

Pure shock waves) 4) G.B. Kistiakowsky, p 951 in Kirk Othmer 5 (1950), pp given in the text (Not included in the 2nd edition) 5) Corner, Ballistics (1950), 100-01 (Corner Noble-Abel equations of state) 6) SAC MS, Ballistics (1951), 18 (Covolume and equation of state of propint gases) 7) Taylor(1952), 34 (Boltzmann and Hirschfelder Roseveare equation of state for the expln products) 69-72 (Rankine-Hugoniot equation of state) 87-98 (Abel, Boltzmann and other equations of state applicable to deton of condensed expls yielding only gaseous products) 114 (Equations of state applicable to deton of condensed expls whose products contain a condensed phase)... 

Baum et al (Ref 8, pp 242-44) showed how the above equation is derived from Abel equation of state, thermodynamic laws 8t Hugoniot equation for ideal gases. They also presented a curve of density-deton velocity relationship for firedamp gas. The curve is nearly a straight line... 

It was stated by Evans Ablow (Ref 66, p 162) Although, as has been observed in previous sections, the shock is curved when the flow diverges, near the axis it is plane. Jones (our Ref 18a) approximated the divergence of the flow near the axis by that in the Prandtl- Meyer expansion around a comer a distance half a diameter (d/2), from the axis. He assumed the Abel equation of state, complete reaction, and the plane form of the C J condition... 

It is std practice in foe formulation of interior ballistic theory to assume an equation of state of the simple covolume type. For a gas obeying the Abel equation of state, the internal energy depends only on temp and not on density. It is expressed as... 

RT = (p+ a/v2) - (v - b), it is the quantity b which corresponds to four times the square root of the space occupied by the molecules themselves (Ref 6). The classical equation of state used by ballisticians and known as the Noble - Abel equation of state is written... 

The major difficulty in applying this hydrodynamic theory of detonation to practical cases lies in the calculation of E2, the specific internal energy of the explosion products immediately behind the detonation front, without which the Rankine-Hugoniot curve cannot be drawn. The calculations require a knowledge of the equation of state of the detonation products and also a full knowledge of the chemical equilibria involved, both at very high temperatures and pressures. The first equation of state used was the Abel equation... 

CHEMKIN REAL-GAS A Fortran Package for Analysis of Thermodynamic Properties and Chemical Kinetics in Nonideal Systems, Schmitt, R. G., Butler, P. B. and French, N. B. The University of Iowa, Iowa City, IA. Report UIME PBB 93-006,1993. A Fortran program (rglib.f and rgin-terp.f) used in connection with CHEMKIN-II that incorporates several real-gas equations of state into kinetic and thermodynamic calculations. The real-gas equations of state provided include the van der Waals, Redlich-Kwong, Soave, Peng-Robinson, Becker-Kistiakowsky-Wilson, and Nobel-Abel. 

Section 1 lists those subjects from "Abel s Equation of State to "Detonating Relays which were either already described in Vols 1,... 

Equations of Jones, of Halford-Kistia-kowsky-Wilson-Brinkley and of Noble-Abel are also covolume equations of state... 

Rankine-Hugoniot equations 181-87 (Equations of state which include among others the following Jones Miller, Lennard-Jones Devonshire, Halford-Kistiakowsky-Wilson, Joffe its modification by Su Chang, Taylor, Kihara Hikita, Travers, Cook, Kistiakowsky-Wilson-Brinkley and Polytropic equations) 194 (Landau-Stanyukovich and Hirschfelder et al equations of state 11) J.F. Roth, Explosiv-stoffe 1958, 50 (Abel sche Zustandsgleichung fur die Detonation) 12) Cook (1958), 37 (General equation of state) 62-3 [Halford-Kistiakowsky-W ilson-Brinkley equation of state, (listed as K-H-W-B equation of state)] ... 

Further in his lecture Dunkle stated that the fourth expression used in hydrothermo-dynamic theory of detonation calculation is the equation of state formulated by Sir F. Abel at the end of the 19th-century. This simple equation, described in Section 3, DETONATION (AND EXPLOSION), EQUATIONS OF STATE IN, is still frequently used It gives fairly accurate results for low loading density condensed expls, but for higher densities,... 

Equations of state (detonation and expin), listing of Abel, Allan, Beattie-Bridgeman, Becker, Becker-Kistiakowsky-Wilson, Benedict-Webb-Rubin, Berthelot, Boltzman, Brinkley-Wilson, Caldirola Paterson, Callendar,... 

Since the combustion gases of powders satisfy Abel s equation of state to a good approximation, it is possible by using the auxiliary... 

The most common equation of state for interior ballistics is that of Nobel-Abel ... 

Equilibria for the reaction N20g(g) + N0(g) + N02(g) have been studied by Beattie and Bell ( ), Verhoek and Daniels (2), and Abel and Proisl (3). The results of Beattie and Bell are the most extensive, but they depend on the early analysis of Giauque and Kemp (4) for the simultaneous equilbrium N20 (g) 2 N02(g). The JANAF analysis of the tetroxide dissociation has been used to recalculate the data of Beattie and Bell. Non-ideality of N20 and NOg was allowed for by use of the equations of state of Giauque and Kemp, while non-ideality of NO and N20g was removed by extrapolation of the equilibrium constants to zero pressure in a manner similar to that of Beattie and Bell. The data of Verhoek and Daniels and of Abel and Proisl have not been recalculated. 2nd and 3rd... 

State" (See item a ). For a detailed description of LS equation, see Ref If Ref 10, p 194 Ref 12a, p 246 and Ref 12b, Chapter 14. Briefly the equation is based on the assumption that decomposition products formed on detonation are, because of their high density, not gases, but rather liquids whose particles are in a state of oscillation. This conditions the course of expansion of the decomposition products. L S replaced R of die Abel equation by a coefficient depending on the number of degrees of molecular freedom. We are not describing here the LS equation because the modification by Zel dovich Kompaneets seems to replace it (See next item... 

The limitation of the prescribed diffusion approach was removed, for an isolated ion-pair, by Abell et al. (1972). They noted the equivalence of the Laplace transform of the diffusion equation in the absence of scavenger (Eq. 7.30) and the steady-state equation in the presence of a scavenger with the initial e-ion distribution appearing as the source term (Eq. 7.29 with dP/dt = 0). Here, the Laplace transform of a function/(t) is defined by... 

Suppose Eq. (6) has a solution with the given asymptotic conditions, which holds true in a wide range of cases [2] then one associates to a given/( ) a solution of the steady state of the Vlasov-Newton equation. There are various restrictions on possible functions/( ) It must be positive or zero and such that the total mass is finite. Of course, as we said, this is not enough to tell what function f E) is to be chosen. Moreover, knowing l>(r), it is possible in principle to find the function/(E) from Eq. (6) by writing the left-hand side as a function of <1) (instead of r). Then there remains to invert an Abel transform to get back/(E). We shall comment now on the impossibility of applying the usual methods of equilibrium statistical mechanics to the present problem (that is, the determination of f E) from a principle of maximization of entropy for instance). 

To find the concentration distribution at the surface in a non-equilibrium state and to check condition (9.13) used in deriving Eq. (9.15), the latter is substituted into the left-hand side of Eq. (8.85). In this case Eq. (8.85) turns ino the Abel integral equation and its solution can be obtained immediately (Dukhin 1965),... 

For a power law material (Sect. 1.6), an interesting method of solution has been developed by Walton et al. (1978). This has no apparent relationship with the method developed here. They derive, for steady-state conditions, an integral equation, which is a generalized Abel-type equation, essentially generalizing (3.2.11), and obtain explicit solutions. Walton (1984) generalized the method to apply to a material for which the shear relaxation function varies with depth also according to a power law. 

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