A Zeldovich spontaneous wave, also referred to as Zeldovich gradient mechanism, is a reaction wave that propagates spontaneously in a reacting medium with a nonuniform initial temperature distribution when there is no interaction between different fluid elements. The concept was put forward by
Yakov Zeldovich
Yakov Borisovich Zeldovich ( be, Я́каў Бары́савіч Зяльдо́віч, russian: Я́ков Бори́сович Зельдо́вич; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet physicist of Bel ...
in 1980, based on his earlier work with his coworkers. The spontaneous wave is different from the other two conventional combustion waves, namely the subsonic
deflagration
Deflagration (Lat: ''de + flagrare'', "to burn down") is subsonic combustion in which a pre-mixed flame propagates through a mixture of fuel and oxidizer. Deflagrations can only occur in pre-mixed fuels. Most fires found in daily life are diffu ...
s and supersonic
detonations
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 ...
. The wave, although strictly speaking unrealistic because
gasdynamic effects are neglected, is often cited to explain the yet-unsolved problem of
deflagration to detonation transition (DDT).
The mechanism behind the spontaneous wave is readily explained by considering a reaction medium at rest with a nonuniform temperature distribution such that the spatial temperature gradients are small or at least it is not sufficiently large (large temperature gradients will evidently lead to interactions between adjacent fluid elements via heat conduction). Corresponding to each fluid element with a definite temperature value, there is an
adiabatic induction period, the time it takes to undergo thermal explosion in the absence of any
heat loss
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
mechanism. Thus, each fluid element will undergo thermal explosion at a definite time as if it is isolated from the rest of the gas. A sequence of these successive
self-ignitions can be identified as some sort of a reaction front and tracked. The spontaneous wave is influenced by the initial condition and is independent of
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
and 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 elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
.
Description of the spontaneous reaction wave
Let
be the initial temperature distribution, which is non trivial, indicating that chemical reactions at different points in space proceed at different rates. To this distribution, we can associate a function
, where
is the adiabatic induction period. Now, define in space some surface
; suppose if
, then this surface for some constant will be parallel to
-plane. Examine the change of position of this surface with the passage of time according to
[Zeldovich, I. A., Barenblatt, G. I., Librovich, V. B., & Makhviladze, G. M. (1985). Mathematical theory of combustion and explosions. page 294-296.]
:
From this, we can easily extract the direction and the propagation speed of the spontaneous front. The direction of the wave is clearly normal to this surface which is given by
and the rate of propagation is just the magnitude of inverse of the gradient of
:
:
Note that adiabatic thermal runaways at different places are not casually connected events and therefore
can assume, in principle, any positive value. By comparing
with other relevant speeds such as, the deflagration speed,
, the
sound speed
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as wel ...
,
and the speed of the
Chapman–Jouguet detonation wave,
, we can identify different regimes:
*When