This chapter adopts the Chapman-Jouguet approach to derive jump conditions across combustion waves propagating in Van der Waals and Noble-Abel gases. The steady one-dimensional balance equations of mass, momentum and energy, assuming different properties of reactants and products, are applied to obtain the main properties of combustion waves, including velocities, Mach numbers, pressures and temperatures, in terms of the covolumes and intermolecular force parameters. In general, the effects of covolumes are more significant than the effects of the intermolecular attraction forces on Hugoniot curves and on properties of combustion waves. However, theoretical results using the Van der Waals equation of state matched more closely the experimental results for detonations of mixtures of propane and diluted air at high initial pressures.
TopIntroduction
Combustion waves are flame fronts which can propagate in solid, liquid, gas or multiphase media with speeds that depend on the composition of the mixture and on initial and boundary conditions. Deflagrations are combustion waves with subsonic speeds relative to the reactants and, in general, present velocities of 0.1 to 1 m/s in standard conditions (100 kPa and 298 K) and are, approximately, isobaric. Detonations are combustion waves with supersonic speeds relative to the reactants and, for standard conditions, present burning front velocities of order of 1500-4000 m/s, detonation Mach numbers 4 < MD < 8 and detonation pressures varying from 2,5 to 6,0 MPa (Nettleton, 1987).
Studies that led to the discovery of detonations in condensed explosives were made by Abel (1869) and in gaseous mixtures by Berthelot and Vielle (1881). However only in the 1950s a comprehensive bibliography of studies on detonations was documented (Bauer et al., 1991; Manson and Dabora, 1993). Current research on detonations involves different research lines defined by the study of a particular phenomenon (Nettleton, 1987; Kuo, 2005; Lee, 2008).
Shortly after the discovery of the detonation phenomenon, Michelson (1893), Chapman (1889) and Jouguet (1905, 1906) proposed a theory to calculate the detonation velocity of an explosive mixture, based on the solutions developed by Rankine (1870) and Hugoniot (1887, 1889) for the shock wave problem. However, Michelson’s publications were unknown outside Russia and the detonation theory became known only as the CJ theory (Dremin, 1999).
The classical Chapman-Jouguet (CJ) theory considers a combustion wave as a discontinuity with an infinity reaction rate. A steady one-dimensional flow process is assumed, with chemical reactions proceeding instantaneously just after a shock wave passes through the medium. Based on the CJ theory, it is possible to calculate the properties of the combustion wave and to verify that the combustion products attain a minimum velocity satisfying the conservation equations.
Solution of continuity and momentum equations yields the Rayleigh line that relates pressures and specific volumes across the combustion wave. Solution of the energy, continuity, momentum and equation of state is known as the Rankine-Hugoniot relation. In the case of constant thermodynamic properties, the Rankine-Hugoniot curve takes the form of a hyperbola yielding the equilibrium states of the combustion products, given the initial conditions of the reactant mixture and the heat release from the reaction.
According to the CJ approach, the solution for the combustion wave propagation problem corresponds to the case where the Rayleigh line is tangent to the Rankine-Hugoniot curve. Two points are determined, known as upper CJ point and lower CJ point which are solutions, respectively, for detonation and deflagration waves. The CJ theory does not require the knowledge of the combustion wave structure and reaction kinetics, but only the determination of the thermodynamic equilibrium state of the combustion products.