Zeldovich–Liñán–Dold model

In 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 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 in 1948,^[1] later analysed by Liñán using activation energy asymptotics in 1971^[2] and later refined by John W. Dold in the 2000s.^[3][4] The ZLD mechanism mechanism reads as

${\displaystyle {\begin{aligned}{\rm {Branching\,(I):}}&\quad {\rm {{F}+{\rm {{Z}\rightarrow 2{\rm {Z}}}}}}\\{\rm {Recombination\,(II):}}&\quad {\rm {{Z}+{\rm {{M}\rightarrow {\rm {{P}+{\rm {{M}+{\rm {Heat}}}}}}}}}}\end{aligned}}}$

where ${\displaystyle {\rm {F}}}$ is the fuel, ${\displaystyle {\rm {Z}}}$ is an intermediate radical, ${\displaystyle {\rm {M}}}$ is the third body and ${\displaystyle {\rm {P}}}$ 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 readsas

${\displaystyle {\begin{aligned}{\rm {Branching\,(I):}}&\quad {\rm {{F}+{\rm {{Z}\rightarrow 2{\rm {Z}}}}}}\\{\rm {Recombination\,(II):}}&\quad {\rm {{Z}+{\rm {{Z}+{\rm {{M}\rightarrow 2{\rm {{P}+{\rm {{M}+{\rm {{Heat}.}}}}}}}}}}}}\end{aligned}}}$

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.^[5][6][7] Therefore, the rate constants are written as^[8]

${\displaystyle k_{\rm {I}}=A_{\rm {I}}e^{-E_{\rm {I}}/RT},\quad k_{\rm {II}}=A_{\rm {II}}}$

where ${\displaystyle A_{\rm {I}}}$ and ${\displaystyle A_{\rm {II}}}$ are the pre-exponential factors, ${\displaystyle E_{\rm {I}}}$ is the activation energy for chain-branching reaction which is much larger than the thermal energy and ${\displaystyle T}$ 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^[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 ${\displaystyle C_{\mathrm {Z} }(t)}$ of the radical in the ZLD model evolves according to

${\displaystyle {\frac {dC_{\mathrm {Z} }}{dt}}=C_{\mathrm {Z} }\left(A_{\mathrm {I} }C_{\mathrm {F} }e^{-E_{\mathrm {I} }/RT}-A_{\mathrm {II} }\right).}$

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 ${\displaystyle C_{\mathrm {Z} }(0)=0}$ is unstable if the right-side term is positive. If ${\displaystyle C_{\mathrm {F} }(0)=C_{\mathrm {F} ,0}}$ denotes the initial fuel concentration, a crossover temperature ${\displaystyle T^{*}}$ as a temperature at which the branching and recombination rates are equal can be defined, i.e.,^[7]

${\displaystyle e^{E_{\mathrm {I} }/RT^{*}}={\frac {A_{\mathrm {I} }}{A_{\mathrm {II} }}}C_{\mathrm {F} ,0}.}$

When ${\displaystyle T>T^{*}}$, branching dominates over recombination and therefore the radial concentration will grow in time, whereas if ${\displaystyle T<T^{*}}$, 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 ${\displaystyle e^{E_{\mathrm {I} }/RT^{*}}=(A_{\mathrm {I} }/A_{\mathrm {II} })C_{\mathrm {F} ,0}C_{\mathrm {Z} }(0)}$, but since ${\displaystyle C_{\mathrm {Z} }(0)}$ 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.^[9] These regimes exist in both aforementioned models.

Let us consider a premixed flame in the ZLD model. Based on the thermal diffusivity ${\displaystyle D_{T}}$ and the flame burning speed ${\displaystyle S_{L}}$, one can define the flame thickness (or the thermal thickness) as ${\displaystyle \delta _{L}=D_{T}/S_{L}}$. Since the activation energy of the branching is much greater than thermal energy, the characteristic thickness ${\displaystyle \delta _{B}}$ of the branching layer will be ${\displaystyle \delta _{B}/\delta _{L}\sim O(1/\beta )}$, where ${\displaystyle \beta }$ is the Zeldovich number based on ${\displaystyle E_{\mathrm {I} }}$. The recombination reaction does not have the activation energy and its thickness ${\displaystyle \delta _{R}}$ will characterised by its Damköhler number ${\displaystyle Da_{\mathrm {II} }=(D_{T}/S_{L}^{2})/(W_{\mathrm {Z} }A_{\mathrm {II} }^{-1})}$, where ${\displaystyle W_{\mathrm {Z} }}$ is the molecular weight of the intermediate species. Specifically, from a diffusive-reactive balance, we obtain ${\displaystyle \delta _{R}/\delta _{L}\sim O(Da_{\mathrm {II} }^{-1/2})}$ (in the ZL model, this would have been ${\displaystyle \delta _{R}/\delta _{L}\sim O(Da_{\mathrm {II} }^{-1/3})}$).

By comparing the thicknesses of the different layers, the three regimes are classified:^[9]

• Slow Recombination Regime (SRR): ${\displaystyle Da_{\mathrm {II} }=O(1)}$ and ${\displaystyle O(\delta _{B})\ll O(\delta _{R})=O(\delta _{L}).}$
• Intermediate Recombination Regime (IRR): ${\displaystyle Da_{\mathrm {II} }=O(\beta )}$ and ${\displaystyle O(\delta _{B})\ll O(\delta _{R})\ll O(\delta _{L}).}$
• Fast Recombination Regime (FRR): ${\displaystyle Da_{\mathrm {II} }=O(\beta ^{2})}$ and ${\displaystyle O(\delta _{B})=O(\delta _{R})\ll O(\delta _{L}).}$

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 ${\displaystyle \Delta \equiv Da_{\mathrm {II} }/\beta ^{2}}$ becomes large, the critical condition is reached,^[9]

${\displaystyle r=e\left(1+{\frac {0.4162}{\sqrt {\Delta }}}\right)}$

where

${\displaystyle r={\frac {Da_{\mathrm {I} }}{\beta Da_{\mathrm {II} }}}e^{-\beta (1+q)/q},\quad Da_{\mathrm {I} }={\frac {D_{T}/S_{L}^{2}}{(A_{\mathrm {I} }Y_{\mathrm {F} ,0}/W_{\mathrm {F} })^{-1}}}.}$

Here ${\displaystyle q}$ is the heat release parameter, ${\displaystyle Y_{\mathrm {F} ,0}}$ is the unburnt fuel mass fraction and ${\displaystyle W_{\mathrm {F} }}$ is the molecular weight of the fuel.

See also

• Zel'dovich mechanism
• Peters four-step chemistry