Figure 5.15 shows the pattern made by the normal reflection of a detonation on a glass plate coated lightly with carbon soot, which may be from either a wooden match or a kerosene lamp. The cellular structure of the detonation front is quite evident. If a similarly soot-coated polished metal (or Mylar) foil is inserted into a detonation tube, the passage of the detonation wave will leave a characteristic “fish-scale” pattern on the smoked foil. Figure 5.16 is a sequence of laser-Schlieren records of a detonation wave propagating in a rectangular tube. One of the side windows has been coated with smoke, and the fish-scale pattern formed by the propagating detonation front itself is illustrated by the interferogram shown in Figure 5.17. The direction of propagation of the detonation is toward the right. As can be seen in the sketch at the top left corner, there are two triple points. At the first triple point A, AI and AM represent the incident shock and Mach stem of the leading front, while AB is the reflected shock. Point B is the second triple point of another three-shock Mach configuration on the reflected shock AB: The entire shock pattern represents what is generally referred to as a double Mach reflection. The hatched lines denote the reaction front, while the dash-dot lines represent the shear discontinuities or slip lines associated with the triple-shock Mach configurations. The entire front ABCDE is generally referred to as the transverse wave, and it propagates normal to the direction of the detonation motion (down in the present case) at about the sound speed of the hot product gases. It has been shown conclusively that it is the triple-point regions at A and B that “write” on the smoke foil. The exact mechanics of how the triple-point region does the writing is not clear. It has been postulated that the high shear at the slip discontinuity causes the soot particles to be erased.

Figure 5.17 shows a schematic of the motion of the detonation front. The fish-scale pattern is a record of the trajectories of the triple points. It is important to note the cyclic motion of the detonation front. Starting at the apex of the cell at A, the detonation shock front is highly overdriven, propagating at about 1.6 times the equilibrium C−J detonation velocity. Toward the end of the cell at D, the shock has decayed to about 0.6 times the C−J velocity before it is impulsively accelerated back to its highly overdriven state, when the transverse waves collide to start the next cycle again. For the first half of the propagation from A to BC, the wave serves as the Mach stem to the incident shocks of the adjacent cells. During the second half from BC to D, the wave then becomes the incident shock to the Mach stems of the neighboring cells. Details of the variation of the shock strength and chemical reactions inside a cell can be found in a paper by Libouton et al. [20]. AD is usually defined as the length L_c of the cell, and BC denotes the cell diameter (also referred to as the cell width or the transverse-wave spacing). The average velocity of the wave is close to the equilibrium C−J velocity.

The observed motion of a real detonation front is far from the steady and 1D motion given by the ZND model. Instead, it proceeds in a cyclic manner in which the shock velocity fluctuates within a cell about the equilibrium C−J value. Chemical reactions are essentially complete within a cycle or a cell length. However, the gas dynamic flow structure is highly 3D; and full equilibration of the transverse shocks, so that the flow becomes essentially 1D, will probably take an additional distance of the order of a few more cell lengths.

From either the cellular end-on or the axial fish-scale smoke foil, the average cell size λ can be measured. The end-on record gives the cellular pattern at one precise instant. The axial record, however, permits the detonation to be observed as it travels along the length of the foil. It is much easier by far to pick out the characteristic cell size λ from the axial record; thus, the end-on pattern is not used, in general, for cell-size measurements.

Early measurements of the cell size have been carried out mostly in low-pressure fuel-oxygen mixtures diluted with inert gases such as He, Ar, and N₂ [21]. The purpose of these investigations is to explore the details of the detonation structure and to find out the factors that control it. It was not until very recently that Bull et al. [22] made some cell-size measurements in stoichiometric fuel air mixtures at atmospheric pressure. Due to the fundamental importance of the cell size in the correlation with the other dynamic parameters, a systematic program was carried out by Kynstantas to measure the cell size of atmospheric fuel air detonations in all the common fuels (e.g., H₂, C₂H₂, C₂H₄, C₃H₆, C₂H₆, C₃H₈, C₄H₁₀, and the welding fuel MAPP) over the entire range of fuel composition between the limits [23]. Stoichiometric mixtures of these fuels with pure oxygen, and with varying degrees of N₂ dilution at atmospheric pressures, were also studied [24]. To investigate the pressure dependence, Kynstantas et al. [24] also measured the cell size in a variety of stoichiometric fuel oxygen mixtures at initial pressures 10 ≤ P₀ ≤ 200 torr. The minimum cell size usually occurs at about the most detonable composition (ϕ = 1). The cell size λ is representative of the sensitivity of the mixture. Thus, in descending order of sensitivity, C₂H₂, H₂, C₂H₄, and the alkanes C₃H₈, C₂H₆, and C₄H₁₀ are obtained. Methane (CH₄), although belonging to the same alkane family, is exceptionally insensitive to detonation, with an estimated cell size λ ≈ 33 cm for stoichiometric composition as compared with the corresponding value of λ ≈ 5.35 cm for the other alkanes. That the cell size λ is proportional to the induction time of the mixture had been suggested by Shchelkin and Troshin [25] very early. However, to compute an induction time requires that the model for the detonation structure be known, and no universally accepted theory exists as yet for the real 3D structure.

Nevertheless, one can use the classical ZND model for the structure and compute an induction time or, equivalently, an induction-zone length l. While this is not expected to correspond to the cell size λ (or cell length L_c), it may elucidate the dependence of λ on l itself (e.g., a linear dependence λ = Al, as suggested by Shchelkin and Troshin). Westbrook [26,27] has made computations of the induction-zone length l using the ZND model, but his calculations are based on a constant-volume process after the shock, rather than integration along the Rayleigh line. Very detailed kinetics of the oxidation processes are employed. By matching with one experimental point, the proportionality constant A can be obtained. The constant A differs for different gas mixtures (e.g., A = 10.14 for C₂H₄, A = 52.23 for H₂); thus, the 3D gas dynamic processes cannot be represented by a single constant alone over a range of fuel composition for all the mixtures. The chemical reactions in a detonation wave are strongly coupled to the details of the transient gas dynamic processes, with the end product of the coupling being manifested by a characteristic chemical length scale λ (or equivalently L_c) or time scale t_c = l/C₁ (where C₁ denotes the sound speed in the product gases, which is approximately the velocity of the transverse waves) that describes the global rate of the chemical reactions. Since λ≃0.6Lc and C1≃D is the C−J detonation velocity, one has 0.5D, where τc,≃Lc/D, which corresponds to the fact that the chemical reactions are essentially completed within one cell length (or one cycle).

Rarefaction waves are generated circumferentially at the tube as the detonation leaves; then they propagate toward the tube axis, cool the shock-heated gases and, consequently, increase the reaction induction time. This induced delay decouples the reaction zone from the shock, and a deflagration persists. The tube diameter must be large enough so that a core near the tube axis is not quenched, and this core can support the development of a spherical detonation wave.

Some analytical and experimental estimates show that the critical tube diameter is 13 times the detonation cell size (d_c ≥ 13λ) [6]. This result is extremely useful in that only laboratory tube measurements are necessary to obtain an estimate of d_c. It is a value, however, that could change somewhat as more measurements are made.

As in the case of deflagrations, a quenching distance exists for detonations; that is, a detonation will not propagate in a tube diameter below a certain size or between infinitely large parallel plates with separation distance below a certain size. This quenching diameter or distance appears to be associated with the boundary layer growth in the retainer configuration [5]. According to Williams [5], the boundary layer growth has the effect of an area expansion on the reaction zone that tends to reduce the Mach number in the burned gases, so the quenching distance arises from the competition of this effect with the heat release that increases this Mach number. For the detonation to exist, the heat release effect must exceed the expansion effect at the C−J plane; otherwise, the subsonic Mach number and the associated temperature and reaction rate will decrease further behind the shock front, and the system will not be able to recover to reach the C−J state. The quenching distance is that at which the two effects are equal. This concept leads to the relation [5]

The quenching distance detonation limit comes about if the induction period or reaction zone length increases greatly as one proceeds away from the stoichiometric mixture ratio. Then the variation of δ∗ or l will be so great that, no matter how large the containing distance, the quenching condition will be achieved for the given mixture ratio. This mixture is the detonation limit.

Belles [29] essentially established a pure chemical-kinetic-thermodynamic approach to estimating detonation limits. Questions have been raised about the approach, but the line of reasoning developed is worth considering. It is a fine example of coordinating various fundamental elements discussed to this point to obtain an estimate of a complex phenomenon.

Belles' prediction of the limits of detonability takes the following course and deals with the hydrogen/oxygen case. Initially, the chemical kinetic conditions for branched-chain explosion in this system are defined in terms of the temperature, pressure, and mixture composition. The standard shock wave equations are used to express, for a given mixture, the temperature and pressure of the shocked gas before reaction is established (condition 1′). The shock Mach number (M) is determined from the detonation velocity. These results are then combined with the explosion condition in terms of M and the mixture composition to specify the critical shock strengths for explosion. The mixtures are then examined to determine whether they can support the shock strength necessary for explosion. Some cannot, and these define the limit.

The set of reactions that determines the explosion condition of the hydrogen oxygen system is essentially

Consequently, the criterion for explosion is

Using rate constants for k₂ and k₄ and expressing the third-body concentration (M′) in terms of the temperature and pressure by means of the gas law, Belles rewrites Eqn (5.49) in the form

This expression gives a weighting for the effectiveness of other species as third bodies, as compared to H₂ as a third body. Equation (5.50) is then written as a logarithmic expression,

This equation suggests that if a given hydrogen/oxygen mixture, which could have a characteristic value of f_x dependent on the mixture composition, is raised to a temperature and pressure that satisfy the equation, then the mixture will be explosive.

For the detonation waves, the following relationships for the temperature and pressure can be written for the condition (1′) behind the shock front. It is these conditions that initiate the deflagration state in the detonation wave:

From Eqn (5.52), it is apparent that many combinations of pressure and temperature will satisfy the explosive condition. However, if the condition is specified that the ignition of the deflagration state must come from the shock wave, Belles argues that only one Mach number will satisfy the explosive condition. This Mach number, called the critical Mach number, is found by substituting Eqns (5.53) and (5.54) into Eqn (5.52) to give

This equation is most readily solved by plotting the left-hand side as a function of M for the initial conditions. The logarithm term on the right-hand side is calculated for the initial mixture, and M is found from the plot.

The final criterion that establishes the detonation limits is imposed by energy considerations. The shock provides the mechanism whereby the combustion process is continuously sustained; however, the energy to drive the shock—that is, to heat up the unburned gas mixture—comes from the ultimate energy release in the combustion process. But if the enthalpy increase across the shock that corresponds to the critical Mach number is greater than the heat of combustion, an impossible situation arises. No explosive condition can be reached, and the detonation cannot propagate. Thus, the criterion for the detonation of a mixture is

The plot of Δh_c and Δh_s for the hydrogen oxygen case as given by Belles is shown in Figure 5.18. Where the curves cross in Figure 5.18, Δh_c = Δh_s, and the limits are specified. The comparisons with experimental data are very good, as is shown in Table 5.6.

Questions have been raised about this approach to calculating detonation limits, and some believe that the general agreement between experiments and the theory as shown in Table 5.6 is fortuitous. One of the criticisms is that a given Mach number specifies a particular temperature and a pressure behind the shock. The expression representing the explosive condition also specifies a particular pressure and temperature. It is unlikely that there would be a direct correspondence of the two conditions from the different shock and explosion relationships. Equation (5.55) must give a unique result for the initial conditions because of the manner in which it was developed.

Detonation limits have been measured for various fuel oxidizer mixtures. These values and comparison with the deflagration (flammability) limits are given in Table 5.7. Detonation limits are always narrower than the deflagration limits. But for H₂ and the hydrocarbons, one should recall that, because of the product molecular weight, the detonation velocity has its maximum near the rich limit. The deflagration velocity maximum is always very near the stoichiometric value and indeed has its minimum values at the limits. Indeed, the experimental definition of the deflagration limits would require this result.