Combustion-wave structure equations

To obtain a rough physical understanding of the mechanism of the instability, attention may be focused first on a planar detonation subjected to a one-dimensional, time-dependent perturbation. Since the instability depends on the wave structure, a model for the steady detonation structure is needed to proceed with a stability analysis. As the simplest structure model, assume that properties remain constant at their Neumann-spike values for an induction distance after which all of the heat of combustion is released instantaneously. If v is the gas velocity with respect to the shock at the Neumann condition, then may be expressed approximately in terms of the explosion time given by equation (B-57) as Z = vt. From normal-shock relations for an ideal gas with constant specific heats in the strong-shock limit, the Neumann-state conditions are expressible by v/vq = po/p —... 

Because it is difficult to account for changes in the properties of the reaction medium (e.g., permeability, thermal conductivity, specific heat) due to structural transformations in the combustion wave, the models typically assume that these parameters are constant (Aldushin etai, 1976b Aldushin, 1988). In addition, the gas flow is generally described by Darcy s law. Convective heat transfer due to gas flow is accounted for by an effective thermal conductivity coefficient for the medium, that is, quasihomogeneous approximation. Finally, the reaction conditions typically associated with the SHS process (7 2(XX) K and p<10 MPa) allow the use of ideal gas law as the equation of state. 

In this type of representation, all combustion events are collapsed into a discontinuity (the wave). Thus, the unknowns are uu u2, p2, T2, and P2. Since there are four equations and five unknowns, an eigenvalue cannot be obtained. Experimentally it is found that the detonation velocity is uniquely constant for a given mixture. In order to determine all unknowns, one must know something about the internal structure (rate of reaction), or one must obtain another necessary condition, which is the case for the detonation velocity determination. 

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