Since the discussion of the detonation phenomena to be considered here will deal extensively with premixed combustible gases, it is appropriate to introduce much of the material by comparison with deflagration phenomena. As the data in Table 5.1 indicate, deflagration speeds are orders of magnitude less than those of detonation. The simple solution for laminar flame speeds given in Chapter 4 was essentially obtained by starting with the integrated conservation and state equations. However, by establishing the Hugoniot relations and developing a Hugoniot plot, it was shown that deflagration waves are in the very low Mach number regime; then it was assumed that the momentum equation degenerates, and the situation through the wave is one of uniform pressure. The degeneration of the momentum equation ensures that the wave velocity to be obtained from the integrated equations used will be small. To obtain a deflagration solution, it was necessary to have some knowledge of the wave structure and the chemical reaction rates that affected this structure.

As will be shown, the steady solution for the detonation velocity does not involve any knowledge of the structure of the wave. The Hugoniot plot discussed in Chapter 4 established that detonation is a large Mach number phenomenon. It is apparent, then, that the integrated momentum equation is included in obtaining a solution for the detonation velocity. However, it was also noted that there are four integrated conservation and state equations and five unknowns. Thus, other considerations were necessary to solve for the velocity. Concepts proposed by Chapman [7] and Jouguet [8] provided the additional insights that permitted the mathematical solution to the detonation velocity problem. The solution from the integrated conservation equations is obtained by assuming the detonation wave to be steady, planar, and one-dimensional (1D); this approach is called Chapman Jouguet theory. Chapman and Jouguet established, for these conditions, that the flow behind the supersonic detonation is sonic. The point on the Hugoniot curve that represents this condition is called the Chapman Jouguet point, and the other physical conditions of this state are called the Chapman Jouguet conditions. What is unusual about the Chapman Jouguet solution is that, unlike the deflagration problem, it requires no knowledge of the structure of the detonation wave, and equilibrium thermodynamic calculations for the Chapman Jouguet state suffice. As will be shown, the detonation velocities that result from this theory agree very well with experimental observations, even in near-limit conditions when the flow structure near the flame front is highly three-dimensional (3D) [6].

Reasonable models for the detonation wave structure have been presented by Zeldovich [9], von Neumann [10], and Döring [11]. Essentially, they constructed the detonation wave to be a planar shock followed by a reaction zone initiated after an induction delay. This structure, which is generally referred to as the ZND model, will be discussed further in a later section.

As in consideration of deflagration phenomena, other parameters are of import in detonation research. These parameters—detonation limits, initiation energy, critical tube diameter, quenching diameter, and thickness of the supporting reaction zone—require a knowledge of the wave structure and hence of chemical reaction rates. Lee [6] refers to these parameters as “dynamic” to distinguish them from the equilibrium “static” detonation states, which permit the calculation of the detonation velocity by Chapman Jouguet theory.

Calculations of the dynamic parameters using a ZND wave structure model do not agree with experimental measurements, mainly because the ZND structure is unstable and is never observed experimentally except under transient conditions. This disagreement is not surprising, as numerous experimental observations show that all self-sustained detonations have a 3D cell structure that comes about because reacting blast “wavelets” collide with each other to form a series of waves transverse to the direction of propagation. Currently, there are no universally accepted theories that define this 3D cell structure.

The following section deals with the calculation of the detonation velocity based on Chapman Jouguet theory. The subsequent section discusses the ZND model in detail, and the last deals with the dynamic detonation parameters.