Chapman–Jouguet Condition
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The Chapman–Jouguet condition holds approximately in
detonation Detonation () is a type of combustion involving a supersonic exothermic front accelerating through a medium that eventually drives a shock front propagating directly in front of it. Detonations propagate supersonically through shock waves with ...
waves in
high explosive An explosive (or explosive material) is a reactive substance that contains a great amount of potential energy that can produce an explosion if released suddenly, usually accompanied by the production of light, heat, sound, and pressure. An exp ...
s. It states that the detonation propagates at a
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
at which the reacting gases just reach sonic velocity (in the frame of the leading
shock wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
) as the reaction ceases. David Chapman and Émile Jouguet 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,
John von Neumann John von Neumann ( ; ; December 28, 1903 â€“ February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
, and Werner Döring (the so-called ZND detonation model). In more detail (in the ZND model) in the frame of the leading shock of the detonation wave, gases enter at supersonic velocity and are compressed through the shock to a high-density, subsonic flow. This sudden change in pressure initiates the chemical (or sometimes, as in
steam explosion A steam explosion is an explosion caused by violent boiling or flashing of water or ice into steam, occurring when water or ice is either superheated, rapidly heated by fine hot debris produced within it, or heated by the interaction of molten ...
s, 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
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
s 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 Rarefaction is the reduction of an item's density, the opposite of compression. Like compression, which can travel in waves (sound waves, for instance), rarefaction waves also exist in nature. A common rarefaction wave is the area of low relati ...
region beyond the CJ plane. This simple one-dimensional model is quite successful in explaining detonations. However, observations of the structure of real chemical detonations show a complex three-dimensional structure, with parts of the wave traveling faster than average, and others slower. Indeed, such waves are quenched as their structure is destroyed. The Wood–Kirkwood detonation theory can correct for some of these limitations.


Mathematical description

Source:Williams, F. A. (2018). Combustion theory. CRC Press. The Rayleigh line equation and the Hugoniot curve equation obtained from the Rankine–Hugoniot relations for an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
, with the assumption of constant specific heat and constant molecular weight, respectively are :\begin \frac &= -\mu \\ \tilde &= \frac, \end where \gamma is the
specific heat ratio In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volu ...
and : \tilde p = \frac,\quad \tilde v = \frac, \quad \alpha = \frac, \quad \mu = \frac. Here the subscript 1 and 2 identifies flow properties (pressure p, density \rho) upstream and downstream of the wave and m is the constant mass flux and q is the heat released in the wave. The slopes of Rayleigh line and Hugoniot curve are :\begin \frac &= \frac, \\ pt \frac &= -\frac. \endâ‹… At the Chapman-Jouguet point, both slopes are equal, leading the condition that :\tilde p = \frac. Substituting this back into the Rayleigh equation, we find :\mu = \gamma\frac. Using the definition of mass flux m \equiv \rho_1 u_1 = \rho_2 u_2, where u denotes the flow velocity, we find :M_2 = \frac = 1 where M is the
Mach number The Mach number (M or Ma), often only Mach, (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Austrian physicist and philosopher Erns ...
and c is the
speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elasticity (solid mechanics), elastic medium. More simply, the speed of sound is how fast vibrations travel. At , the speed of sound in a ...
, in other words, downstream flow is sonic with respect to the Chapman-Jouguet wave. Explicit expression for the variables can be derived, :\begin \tilde_\pm &= 1 + \alpha(\gamma - 1)\left\, \\ \tilde v_\pm &= 1 + \frac\left\, \\ \mu_\pm &= \gamma + \alpha(\gamma^2 - 1)\left\. \end The upper sign applies for the Upper Chapman-Jouguet point (
detonation Detonation () is a type of combustion involving a supersonic exothermic front accelerating through a medium that eventually drives a shock front propagating directly in front of it. Detonations propagate supersonically through shock waves with ...
) and the lower sign applies for the Lower Chapman-Jouguet point (
deflagration Deflagration (Lat: ''de + flagrare'', 'to burn down') is subsonic combustion in which a pre-mixed flame propagates through an explosive or a mixture of fuel and oxidizer. Deflagrations in high and low explosives or fuel–oxidizer mixtures ma ...
). Similarly, the upstream Mach number can be found from :M_ = \left + \frac\right\frac \pm \left frac\right\frac and the temperature ratio \tilde = T_2/T_1 can be found from the relation \tilde = \tilde\tilde.


See also

* Taylor–von Neumann–Sedov blast wave * Zeldovich–Taylor flow


References


Further reading

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