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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 :\begin \rm & \quad \rm + \rm \rightarrow 2\rm \\ \rm & \quad \rm + \rm \rightarrow \rm +\rm +\rm \end where \rm 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 ...
, \rm 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 ...
, \rm is the third body and \rm 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 :\begin \rm & \quad \rm + \rm \rightarrow 2\rm \\ \rm & \quad\rm + \rm + \rm \rightarrow 2\rm +\rm +\rm. \end 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 asLee, S. R., & Kim, J. S. (2024). The Asymptotic Structure of Strained Chain-Branching Premixed Flames Under Nonadiabatic Conditions. Combustion Science and Technology, 1-27. :k_ = A_ e^, \quad k_ = A_ where A_ and A_ are the pre-exponential factors, E_ is the activation energy for chain-branching reaction which is much larger than the thermal energy and T 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 C_(t) of the radical in the ZLD model evolves according to :\frac = C_\left(A_C_e^ - A_\right). 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 C_(0)=0 is unstable if the right-side term is positive. If C_(0)=C_ denotes the initial fuel concentration, a ''crossover temperature'' T^* as a temperature at which the branching and recombination rates are equal can be defined, i.e., :e^ = \frac C_. When T>T^*, branching dominates over recombination and therefore the radial concentration will grow in time, whereas if T, 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. In the ZL model, one would have obtained e^ = (A_/A_) C_ C_(0), but since C_(0) is zero or vanishingly small in the perturbed state, there is no crossover temperature.


Three regimes

In his analysis, Liñán showed that there exists three types of regimes, namely, ''slow recombination regime'', ''intermediate recombination regime'' and ''fast recombination regime''.Lee, S. R., & Kim, J. S. (2024). The asymptotic solution of near-limit chain-branching premixed flames with the Zel’dovich–Liñán two-step mechanism in the linear and fast recombination regime. Combustion and Flame, 265, 113441. These regimes exist in both aforementioned models. Let us consider a premixed flame in the ZLD model. Based on the thermal diffusivity D_T and the flame burning speed S_L, one can define the flame thickness (or the thermal thickness) as \delta_L=D_T/S_L. Since the activation energy of the branching is much greater than thermal energy, the characteristic thickness \delta_B of the branching layer will be \delta_B/\delta_L \sim O(1/\beta), where \beta is the
Zeldovich number The Zeldovich number is a dimensionless number which provides a quantitative measure for the activation energy of a chemical reaction which appears in the Arrhenius exponent, named after the Russian scientist Yakov Borisovich Zeldovich, who along ...
based on E_. The recombination reaction does not have the activation energy and its thickness \delta_R will characterised by its Damköhler number Da_=(D_T/S_L^2)/(W_ A_^), where W_ is the
molecular weight A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
of the intermediate species. Specifically, from a diffusive-reactive balance, we obtain \delta_R/\delta_L \sim O(Da_^) (in the ZL model, this would have been \delta_R/\delta_L \sim O(Da_^)). By comparing the thicknesses of the different layers, the three regimes are classified: The fast recombination represents situations near the flammability limits. As can be seen, the recombination layer becomes comparable to the branching layer. 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 \Delta \equiv Da_/\beta^2 becomes large, the critical condition is reached, :r=e\left(1+ \frac\right) where :r = \frac e^, \quad Da_ = \frac. Here q is the heat release parameter, Y_ is the unburnt fuel mass fraction and W_ is the molecular weight of the fuel.


See also

* Zel'dovich mechanism * Peters four-step chemistry


References

{{DEFAULTSORT:Zeldovich-Linan model Chemical kinetics Combustion Reaction mechanisms Chemical reactions