In combustion, Zeldovich–Liñán model is a two-step reaction model for the combustion processes, named after Yakov Borisovich Zeldovich and Amable Liñán.
The model includes a chain-branching and a chain-breaking (or radical recombination) reaction.
The model was first introduced by Zeldovich in 1948[1] and later analysed by Liñán using activation energy asymptotics in 1971.
[2] The mechanism with a quadratic or second-order recombination that were originally studied reads as where
The mechanism with a linear or first-order recombination is known as Zeldovich–Liñán–Dold model which was introduced by John W.
[3][4] This mechanism reads as In both models, the first reaction is the chain-branching reaction (it produces two radicals by consuming one radical), which is considered to be auto-catalytic (consumes no heat and releases no heat), with very large activation energy and the second reaction is the chain-breaking (or radical-recombination) reaction (it consumes radicals), where all of the heat in the combustion is released, with almost negligible activation energy.
First of all, the so-called cold-boundary difficulty in premixed flames does not occur in the ZLD model[4] and secondly the so-called crossover temperature exist in the ZLD, but not in the ZL model.
[9] For simplicity, consider a spatially homogeneous system, then the concentration
of the radical in the ZLD model evolves according to It is clear from this equation that the radical concentration will grow in time if the righthand side term is positive.
More preceisley, the initial equilibrium state
is unstable if the right-side term is positive.
denotes the initial fuel concentration, a crossover temperature
as a temperature at which the branching and recombination rates are equal can be defined, i.e.,[7] When
, branching dominates over recombination and therefore the radial concentration will grow in time, whereas if
, recombination dominates over branching and therefore the radial concentration will disappear in time.
In a more general setup, where the system is non-homogeneous, evaluation of crossover temperature is complicated because of the presence of convective and diffusive transport.
is zero or vanishingly small in the perturbed state, there is no crossover temperature.
[9] These regimes exist in both aforementioned models.
Let us consider a premixed flame in the ZLD model.
Since the activation energy of the branching is much greater than thermal energy, the characteristic thickness
The recombination reaction does not have the activation energy and its thickness
is the molecular weight of the intermediate species.
Specifically, from a diffusive-reactive balance, we obtain
By comparing the thicknesses of the different layers, the three regimes are classified:[9]
The fast recombination represents situations near the flammability limits.
The criticality is achieved when the branching is unable to cope up with the recombination.
Such criticality exists in the ZLD model.
Su-Ryong Lee and Jong S. Kim showed that as
becomes large, the critical condition is reached,[9] where Here
is the unburnt fuel mass fraction and