In
combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustion ...
, Zeldovich–Liñán–Dold model or ZLD model or ZLD mechanism is a two-step reaction model for the combustion processes, named after
Yakov Borisovich Zeldovich,
Amable Liñán
Amable Liñán Martínez (born 1934 in Noceda de Cabrera, Castrillo de Cabrera, León, Spain) is a Spanish aeronautical engineer working in the field of combustion.
Biography
He holds a PhD in Aeronautical Engineering from the Technical Uni ...
and
John W. Dold. The model includes a chain-branching and a chain-breaking (or radical recombination) reaction. The model was first introduced by
Zeldovich
Yakov Borisovich Zeldovich (, ; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet physicist of Belarusian origin, who is known for his prolific contributions in physical cosmology, physics of thermonuclear reactions ...
in 1948, later analysed by
Liñán using
activation energy asymptotics
Activation energy asymptotics (AEA), also known as large activation energy asymptotics, is an asymptotic analysis used in the combustion field utilizing the fact that the reaction rate is extremely sensitive to temperature changes due to the large ...
in 1971 and later refined by
John W. Dold in the 2000s.
[Dold, J. W. (2007). Premixed flames modelled with thermally sensitive intermediate branching kinetics. Combustion Theory and Modelling, 11(6), 909-948.] The ZLD mechanism mechanism reads as
:
where
is the
fuel
A fuel is any material that can be made to react with other substances so that it releases energy as thermal energy or to be used for work (physics), work. The concept was originally applied solely to those materials capable of releasing chem ...
,
is an intermediate
radical
Radical (from Latin: ', root) may refer to:
Politics and ideology Politics
*Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century
*Radical politics ...
,
is the third body and
is the product. This mechanism exhibits a ''linear or first-order recombination''. The model originally studied before Dold's refinement pertains to a ''quadratic or second-order recombination'' and is referred to as Zeldovich–Liñán model. The ZL 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
In the Arrhenius model of reaction rates, activation energy is the minimum amount of energy that must be available to reactants for a chemical reaction to occur. The activation energy (''E''a) of a reaction is measured in kilojoules per mole (k ...
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
In the Arrhenius model of reaction rates, activation energy is the minimum amount of energy that must be available to reactants for a chemical reaction to occur. The activation energy (''E''a) of a reaction is measured in kilojoules per mole (k ...
.
[Dold, J., Daou, J., & Weber, R. (2004). Reactive-diffusive stability of premixed flames with modified Zeldovich-Linán kinetics. Simplicity, Rigor and Relevance in Fluid Mechanics, 47-60.] Therefore, the
rate constant
In chemical kinetics, a reaction rate constant or reaction rate coefficient () is a proportionality constant which quantifies the rate and direction of a chemical reaction by relating it with the concentration of reactants.
For a reaction between ...
s are written as
[Lee, S. R., & Kim, J. S. (2024). The Asymptotic Structure of Strained Chain-Branching Premixed Flames Under Nonadiabatic Conditions. Combustion Science and Technology, 1-27.]
:
where
and
are the pre-exponential factors,
is the activation energy for chain-branching reaction which is much larger than the thermal energy and
is the temperature.
Crossover temperature
Albeit, there are two fundamental aspects that differentiate Zeldovich–Liñán–Dold (ZLD) model from the Zeldovich–Liñán (ZL) model. First of all, the so-called cold-boundary difficulty in premixed flames does not occur in the ZLD model and secondly the so-called crossover temperature exist in the ZLD, but not in the ZL model.
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. If
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.,
:
When
, branching dominates over recombination and therefore the radial concentration will grow in time, whereas if