RANKINE-HUGONIOT RELATIONS

In this chapter we shall classify the various types of infinite, plane, steady-state, one-dimensional flows involving exothermic chemical reactions, in which the properties become uniform as x oo. Such a classification provides a framework within which plane deflagration and detonation waves may be investigated. The experimental conditions under which these waves appear are described in Chapters 5 and 6, where detailed analyses of each type of wave are presented. 

The Rankine-Hugoniot relations are the equations relating the properties on the upstream and downstream sides of these combustion waves. In this chapter, general Rankine-Hugoniot equations are derived and discussed first then the Hugoniot curve for a simplified system is studied in detail in order to delineate explicitly the various burning regimes. 

Since the major changes in the values of the flow variables in these systems usually take place over a very short distance (in nearly all cases less than a few centimeters see Sections 5.1.2 and 6.1), in many problems deflagration and detonation waves can be treated as discontinuities (at which heat addition occurs) in the flow equations for an TdeaT (inviscid, nondiffusive, non-heat-conducting, nonreacting) fluid. In such problems, the equations derived in this chapter provide all of the information that is required concerning these waves, except their speed of propagation. 

Some other texts discussing the Rankine-Hugoniot equations are listed in [l]-[5]. 

If the reactant and product are assumed to be in thermodynamic equilibrium, and e2 are given as known functions of pressure and density, 

The Rankine-Hugoniot equation given by Eq. (3.10) or Eq. (3.9) is shown in Fig. 3-2 as a function of 1 /p and p called the Hugoniot curve. The Hugoniot curve for q= 0, i.e., no chemical reactions, passes the initial point (1/pi, Pi) and is exactly equivalent to the shock wave described in Chapter 1. When heat q is produced in the combustion wave, the Hugoniot curve shifts to the position shown in Fig. 3-2. It is evident that two different types of combustion are possible on the Hugoniot curve (1) a detonation, in which pressure and density increase, and (2) a deflagration, in which pressure and density decrease. 

Equations (3.1) and (3.2) yield the following relationship, the so-called Rayleigh equation  

As shown in Fig. 3-3, the two lines of the tangents from the initial point 1 (l/pi,pi) to the points J and K (l/p2,p2) on the Hugoniot curve represent the Rayleigh lines which are expressed by161 

The mass flow rate given by Eq. (3.1) and the velocity u2 at point J or K are represented by 

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Conservation equations Expressions that equate the mass, momentum, and energy across a steady wave or shock discontinuity ((2.1)-(2.3)). Also known as the jump conditions or the Rankine-Hugoniot relations. 

From the R-H (Rankine-Hugoniot) relations at the shock, there was obtd ... 

Theory. If p is pressure, v - specific volume, e - specific internal energy, D detonation velocity, u - particle velocity, C - sound velocity, y - adiabatic exponent and q -specific.detonation energy, the velocity of propagation and particle velocity immediately behind any plane detonation wave in an explosive, defined by initial conditions, pD, v0, eQ, and uQ, are given by the first two Rankine-Hugoniot relations ... 

Detonation, Rankine-Hugoniot Equations and Rankine-Hugoniot Relation in. See under DETONATION (AND EXPLOSION), THEORIES and in paper of M.W. Evans ... 

Evaluation of the density at the front, together with the Rankine-Hugoniot relations and the measured front velocity, determines the pressure and particle velocity there. In practice, this requires an additional assumption, which will be made throughout. Since the reaction zone is much smaller than the foil spacing, the reaction is treated as instantaneously complete within the shock transition, and the final state to which the Rankine-Hugoniot equations apply is taken to be the equilibrium state at the end of the reaction zone. No evidence of a reaction zone can be detected either in the analysis of the foil data or on the radiographs. 

It is immediately apparent that the energy equation (the Rankine-Hugoniot relation) is expressed entirely in terms of thermodynamic quantities. 

Once a detonation is achieved, the results of the simulations are soon in good quantitative agreement with the Rankine-Hugoniot relations " of continuum theory. These relations state that the following quantities are conserved across a planar shockfront ... 

Rankine-Hugoniot relations are so well obeyed within about 5 nm of the front implies, via a classic argument not repeated here, that the simulated detonation is moving at the minimum velocity consistent with the conservation conditions, as predicted by ZND theory. We return to a discussion of the CJ point and related issues in Sec. 4, where several methods are presented for establishing the existence and position of the CJ point from the simulations. 

These equations relate the undisturbed explosive lying at rest with pressure Pq = 0 and specific volume Vq = to the state behind the detonation front, which is characterized by a pressure P, a specific volume V, and a particle flow velocity u. Both u and the detonation velocity, D, are measured in the reference frame of the undisturbed material. Because Pq and Vq are known, the Rankine-Hugoniot relations are a set of three equations for the four unknowns, u, D, P, and V. The first relation determines u in terms of D, P, and Vi, which leaves two equations with three unknowns. The first of the remaining equations, Eq. (4b) defines the Rayleigh line while Eq. (4c) defines the Hugoniot curve. The problem is formally determined by selecting the solution of Eqs. (4b) and (4c) that corresponds to the minimum value of D for an unsupported detonation. This additional condition is the Chapman—Jouguet hypothesis, which was put on a firmer foundation by Zel dovich. ... 

Determining the Hugoniot from supported piston simulations requires that the detonation profile from the piston face to the front be steady for a given piston velocity, for only then will the Rankine-Hugoniot relations be satisfied near the piston face. We have shown directly from the simulations that this is an excellent assumption so long as the detonation is overdriven, that is so long as Pi and Vi exceed their corresponding values at the CJ point. This is illustrated in Fig. 21, where we show a series... 

Finally, the profiles of Figure 4 can also be analyzed in terms of the Rankine-Hugoniot relations... 

Longitudinal profiles of shock wave properties in acetylene are shown in Figure 7 for a flyer plate impact speed of 16 km/s and a flyer plate thickness of six unit cells. Profiles for various times up to 1.2 ps after impact are depicted. At early times before the appearance of the release wave, when the mass velocity and density profiles are flat behind the shock front, it is possible to derive the parameters necessary for a Hugoniot analysis. As for methane, it is found that the Rankine-Hugoniot relations are satisfied. The Hugoniot parameters for several of the acetylene and methane simulations are collected in Table 1. Note that for a given flyer plate velocity, the temperature in the reaction zone is much higher for acetylene than for methane due to the exothermicity of the polymerization reactions. 

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