Chapman–Jouguet condition

The Chapman–Jouguet condition holds approximately in detonation waves in high explosives.

[1][2] David Chapman[3] and Émile Jouguet[4] originally (c. 1900) stated the condition for an infinitesimally thin detonation.

A physical interpretation of the condition is usually based on the later modelling (c. 1943) by Yakov Borisovich Zel'dovich,[5] John von Neumann,[6] and Werner Döring[7] (the so-called ZND detonation model).

This sudden change in pressure initiates the chemical (or sometimes, as in steam explosions, physical) energy release.

The energy release re-accelerates the flow back to the local speed of sound.

It can be shown fairly simply, from the one-dimensional gas equations for steady flow, that the reaction must cease at the sonic ("CJ") plane, or there would be discontinuously large pressure gradients at that point.

The sonic plane forms a so-called choke point that enables the lead shock, and reaction zone, to travel at a constant velocity, undisturbed by the expansion of gases in the rarefaction region beyond the CJ plane.

[10] Source:[11] The Rayleigh line equation and the Hugoniot curve equation obtained from the Rankine–Hugoniot relations for an ideal gas, with the assumption of constant specific heat and constant molecular weight, respectively are where

is the specific heat ratio and Here the subscript 1 and 2 identifies flow properties (pressure

The slopes of Rayleigh line and Hugoniot curve are At the Chapman-Jouguet point, both slopes are equal, leading the condition that Substituting this back into the Rayleigh equation, we find Using the definition of mass flux

is the speed of sound, in other words, downstream flow is sonic with respect to the Chapman-Jouguet wave.

Explicit expression for the variables can be derived, The upper sign applies for the Upper Chapman-Jouguet point (detonation) and the lower sign applies for the Lower Chapman-Jouguet point (deflagration).

Similarly, the upstream Mach number can be found from and the temperature ratio