Correlations

Cell width can not currently be computed or predicted directly. Current hydrodynamic simulations that attempt to reproduce detonation cellular structure make large sacrifices, usually by using single step, irreversible reactions and constant properties, and are capable of exhibiting qualitatively correct phenomena, but can not be used as engineering predictive tools. Simulations that compute detailed chemical kinetics generally lack spatial resolution such that the reaction zone structure is meaningless.

Our analysis proceeds along dimensional analysis grounds. The ratio of cell width to reaction zone thickness ( $\lambda/\Delta$) is a function of other nondimensional parameters of the flow. For a system characterized by a single reaction with activation energy E_a, energy release q, ratio of specific heats $\gamma$, and detonation Mach number M_CJ, we expect that

$$\frac{\lambda}{\Delta} \approx f\left(M_{\rm CJ}, \gamma, \frac{q}{RT_{vN}}, \frac{E_a}{RT_{vN}}\right)$$

In general, f may include a large number of other parameters, but these are believed to be the most influential. The activation energy and heat release normalization factors include the von Neumann temperature because it is most relevant to the reaction zone behavior. While the general form of this function has not been found, certain useful approximations are possible. For instance, for a given fuel - oxidizer - diluent system at constant equivalence ratio and initial pressure, the function f is generally constant with respect to variation in dilution ratio. A slightly more general approach is illustrated in Figs. 12, 13, 14, and 15. The cell width data from different test conditions are plotted together by using computed reaction zone thickness for each condition as the abscissa.

  

Figure: Correlation of cell width measurements with computed ZND reaction zone thicknesses using the modified [Miller and Bowman (1989)] mechanism for stoichiometric H₂-N₂O mixtures in air and N₂

  

Figure: Correlation of cell width measurements with computed ZND reaction zone thicknesses using the modified [Miller and Bowman (1989)] mechanism for stoichiometric CH₄-O₂ mixtures in N₂

  

Figure: Correlation of cell width measurements with computed ZND reaction zone thicknesses using the modified [Miller and Bowman (1989)] mechanism for stoichiometric CH₄-N₂O mixtures in air and N₂

  

Figure: Correlation of cell width measurements with computed ZND reaction zone thicknesses using the modified [Miller and Bowman (1989)] mechanism for stoichiometric NH₃-O₂ and NH₃-N₂O mixtures in N₂ and air

  

Figure: Correlation of cell width measurements with computed ZND reaction zone thicknesses using the modified [Miller and Bowman (1989)] mechanism for model tank mixtures in air

For each fuel - oxidizer - diluent system, at constant equivalence ratio, cell width is found to obey a power law with respect to ZND reaction zone thickness. Note that while air is presented as a diluent along with N₂ in the H₂-N₂O and CH₄-N₂O mixtures, it acts as an oxidizer also and therefore mixtures with air are unique from N₂ systems. The undiluted H₂-N₂O data are considered to be a subset of the N₂ dilution data but not the air dilution data because O₂ is found to have a significant effect on N₂O mixtures even at very small concentrations.

A power law has been least-squares fit to the available cell width - reaction zone thickness data for each mixture, as shown in Figs. 12, 13, and 14. The data for H₂-N₂O-Air do not correlate well to a power law. The correlations for CH₄-O₂-N₂, CH₄-N₂O-N₂, and CH₄-N₂O-Air are good, although the number of data points is limited in the N₂O mixtures. A number of initial pressures are represented in the literature CH₄-O₂-N₂ data along with various dilution ratios.