When the detonation velocity was calculated in the previous section, the conservation equations were used, and no knowledge of the chemical reaction rate or structure was necessary. The wave was assumed to be a discontinuity. This assumption is satisfactory because these equations placed no restriction on the distance between a shock and the seat of the generating force.

But to look somewhat more closely at the structure of the wave, one must deal with the kinetics of the chemical reaction. The kinetics and mechanism of reaction give the time and spatial separation of the front and the C−J plane.

The distribution of pressure, temperature, and density behind the shock depends on the fraction of material reacted. If the reaction rate is exponentially accelerating (i.e., follows an Arrhenius law and has a relatively large overall activation energy like that normally associated with hydrocarbon oxidation), the fraction reacted changes very little initially; the pressure, density, and temperature profiles are very flat for a distance behind the shock front and then change sharply as the reaction goes to completion at a high rate.

Figure 5.13, a graphical representation of the ZND theory, shows the variation of the important physical parameters as a function of spatial distribution. Plane 1 is the shock front, plane 1′ is the plane immediately after the shock, and plane 2 is the C−J plane. In the previous section, the conditions for plane 2 were calculated and u₁ was obtained. From u₁ and the shock relationships or tables, it is possible to determine the conditions at plane 1′. Typical results are shown in Table 5.5 for various hydrogen and propane detonation conditions. Note from this table that (ρ₂/ρ_l) ≅ 1.8. Therefore, for many situations the approximation that u₁ is 1.8 times the sound speed, c₂, can be used.

Thus, as the gas passes from the shock front to the C−J state, its pressure drops about a factor of 2, the temperature rises about a factor of 2, and the density drops by a factor of 3. It is interesting to follow the model on a Hugoniot plot, as shown in Figure 5.14.

There are two alternative paths by which a mass element passing through the wave from ε = 0 to ε = 1 may satisfy the conservation equations and at the same time change its pressure and density continuously, not discontinuously, with a distance of travel.

The element may enter the wave in the state corresponding to the initial point and move directly to the C−J point. However, this path demands that this reaction occur everywhere along the path. Since there is little compression along this path, there cannot be sufficient temperature to initiate any reaction. Thus, there is no energy release to sustain the wave. If on another path a jump is made to the upper point (1′), the pressure and temperature conditions for initiation of reaction are met. In proceeding from 1 to 1′, the pressure does not follow the points along the shock Hugoniot curve.

The general features of the model in which a shock, or at least a steep pressure and temperature rise, creates conditions for reaction and in which the subsequent energy release causes a drop in pressure and density, have been verified by measurements in a detonation tube [18]. Most of these experiments were measurements of density variation by X-ray absorption. The possible effect of reaction rates on this structure is depicted in Figure 5.14 as well [19].

The ZND concepts consider the structure of the wave to be 1D and are adequate for determining the “static” parameters μ, ρ₂, T₂, and P₂. However, there is now evidence that all self-sustaining detonations have a 3D cellular structure.