Zeldovich–Liñán model
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 is the fuel, is an intermediate radical, is the third body and is the product. The mechanism with a linear or first-order recombination is known as Zeldovich–Liñán–Dold model which was introduced by John W. Dold.[3][4] This mechanism reads as
In both models, the first reaction is the chain-branching reaction, 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, where all of the heat in the combustion is released, with almost negligible activation energy.[5][6][7] Therefore, the rate constants are written as[8]
where and are the pre-exponential factors, is the activation energy for chain-branching reaction which is much larger than than the thermal energy and is the temperature.
In his analysis, Liñán showed that there exists three types of regimes, namely, slow recombination regime, intermediate recombination regime and fast recombination regime.[9] These regimes exist in both aforementioned models.
Crossover temperature[edit]
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[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
}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7008dc4d1b4cc31771b64974a71e47f5996cb711)
![{\displaystyle {\frac {d[\mathrm {Z} ]}{dt}}=[\mathrm {Z} ]\left(A_{\mathrm {I} }[\mathrm {F} ]e^{-E_{\mathrm {I} }/RT}-A_{\mathrm {II} }\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a606e107f616371ef191138a0980162ee5ec889e)
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, i.e.,[7]
=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5de94cc1f25409e7cf87fe65681f307a2506b6c1)
=[\mathrm {F} ]_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dfff62b769c067810d928ba2b6ba48bc7e3dc61)
![{\displaystyle e^{E_{\mathrm {I} }/RT^{*}}={\frac {A_{\mathrm {I} }}{A_{\mathrm {II} }}}[\mathrm {F} ]_{0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a96acb1272c39f6b9e15937bc9f233896dbf90d1)
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.
In the ZL model, one would have obtained , but since is zero or vanishingly small in the perturbed state, there is no crossover temperature.
![{\displaystyle e^{E_{\mathrm {I} }/RT^{*}}=(A_{\mathrm {I} }/A_{\mathrm {II} })[\mathrm {F} ]_{0}[\mathrm {Z} ](0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bab053571a8f4ea62b0599c27730908a8a01986d)
}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8997ff51f417997838f9d9ca3f9ce75b74750e6c)
See also[edit]
References[edit]
- ^ Zeldovich, Y. B. (1948). K teorii rasprostraneniya plameni. Zhurnal Fizicheskoi Khimii, 22(1), 27-48.
- ^ Liñán, A. (1971). A theoretical analysis of premixed flame propagation with an isothermal chain reaction. AFOSR Contract No. E00AR68-0031, 1.
- ^ Dold, J. W., Thatcher, R. W., Omon-Arancibia, A., & Redman, J. (2002). From one-step to chain-branching premixed flame asymptotics. Proceedings of the Combustion Institute, 29(2), 1519-1526.
- ^ a b Dold, J. W. (2007). Premixed flames modelled with thermally sensitive intermediate branching kinetics. Combustion Theory and Modelling, 11(6), 909-948.
- ^ Gubernov, V. V., Kolobov, A. V., Polezhaev, A. A., & Sidhu, H. S. (2011). Pulsating instabilities in the Zeldovich–Liñán model. Journal of mathematical chemistry, 49(5), 1054-1070.
- ^ Tam, R. Y. (1988). Damköhler-number ratio asymptotics of the Zeldovich-Liñán model. Combustion science and technology, 62(4-6), 297-309.
- ^ a b 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.
- ^ 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.
- ^ a b 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.