Rankine - Hugoniot

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)... 

The Rankine-Hugoniot curve is sometimes referred to as the shock adiabat (especially in the Soviet literature). This terminology reflects the fact that the shock process is so fast that there is insufficient time for heat... 

The Hugoniot can be described with a differential equation by taking the total differential of the Rankine-Hugoniot equation (2.4)... 

Since this equation came directly from differentiation of the Rankine-Hugoniot equation, it only holds true on the Hugoniot. We can also write T dS as a total differential in terms of dP and d V... 

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. 

In the simplest case when a single shock state is achieved via a shock front, the Rankine-Hugoniot equations involve six variables U, u, p, Pi, i — Eq, and Pi) thus, measuring three, usually U, m, and p, determines the shock state, pi, , - A- The key assumption underpinning the... 

To demonstrate that the Rayleigh line actually represents the thermodynamic path to which material is subjected on being shocked from state p = 0, F = Fq to P = Pi, F = Fi, we demonstrate below that the shock wave sketched in Fig. 4.1 must be steady. Moreover, the Rankine Hugoniot equations ((4.1)-(4.3)) not only describe the conservation of mass, momentum. 

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... 

Extension of the hydrodynamic theory to explain the variation of detonation velocity with cartridge diameter takes place in two stages. First, the structure of the reaction zone is studied to allow for the fact that the chemical reaction takes place in a finite time secondly, the effect of lateral losses on these reactions is studied. A simplified case neglecting the effects of heat conduction or diffusion and of viscosity is shown in Fig. 2.5. The Rankine-Hugoniot curves for the unreacted explosive and for the detonation products are shown, together with the Raleigh line. In the reaction zone the explosive is suddenly compressed from its initial state at... 

At the shock front in free air, a number of wave properties are interrelated through the Rankine-Hugoniot equations. These three equations are (Reference 5) ... 

The behavior of shock waves is ruled by the Rankine-Hugoniot equations, which express the conditions for conservation of mass, momentum, and energy and can be used to design suitable shock prohles. Referring to the PV diagram (see left panel of Fig. 13), the compressed state (P, V ) can be represented as... 

Though the form of the Rankine-Hugoniot equation, Eqs. (1.42)-(1.44), is obtained when a stationary shock wave is created in a moving coordinate system, the same relationship is obtained for a moving shock wave in a stationary coordinate system. In a stationary coordinate system, the velocity of the moving shock wave is Ml and the particle velocity is given by u = u M2. The ratios of temperature, pressure, and density are the same for both moving and stationary coordinates. 

The basic equations for describing the detonahon characteristics of condensed materials are fundamentally the same as those for gaseous materials described in Sections 3.2 and 3.3. The Rankine-Hugoniot equations used to determine the detonation velocities and pressures of gaseous materials are also used to determine these parameters for explosives. Referring to Sechon 3.2.3, the derivative of the Hugoniot curve is equal to the derivative of the isentropic curve at point J. Then, Eq. (3.13) be-... 

For a transmitted shock wave advancing into any gas at an initial pressure pe of 1 atm, the RH (Rankine-Hugoniot) equation defines a functional relationship between pressure p and particle velocity w behind the wave S3, involving initial pressure, initial specific volume v, and equations of state of the target medium. Similarly, the conditions behind the reflected wave S2 and close to the product-target interface are expressible by means either of the shock wave equations or the Rie-mann adiabatic wave equations in terms of any one such variable and the conditions... 

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