Oscillations of simple exothermic reactions in a closed system. II. Exact Arrhenius kinetics
Oscillations of simple exothermic reactions in a closed system. II. Exact Arrhenius kinetics
复制标题
封闭系统中简单放热反应的振荡。
DOI：
10.1098/rspa.1988.0038
复制
发表时间：
1988
期刊：
影响因子：
--
通讯作者：
S. Scott
中科院分区：
文献类型：
--
作者：
S. R. Kay;S. Scott
封闭化学系统中最简单的热动力学振荡模型仅需要两个一级反应步骤 (0) P → A 速率 = k0p，(1) A → B 速率 = k1(T)a。假设步骤 (0) 是热中性的，其速率常数不依赖于温度（即活化能为零）。步骤（1）是一个放热过程，速率常数 k1 具有阿累尼乌斯温度依赖性 k1 = A1 e–E1/RT。根据中间体 A 的还原浓度 α 和温升 θ 以无量纲形式编写的控制反应速率和能量平衡方程为： dα/dז = μ e-γז ─ kαf(θ) 和 dθ/dז = αf(θ) ─ θ，其中 μ、γ 和 k 是参数，函数 f(θ) 的形式为 f(θ)(θ) = exp [θ/(1 + ∊θ)]。这些方程的所有动力学行为都可以定性地确定，并且可以根据池化学近似 γ = 0 定量地定性地确定。随着 μ 和 k 的变化，从 Hopf 分岔点开始会出现振荡解。这些点被分析地定位，并导出附加表达式来计算当系统远离这些点时幅度和周期的增长。只要 0 < ∊ < 2/9，就可以找到稳定和不稳定的极限环（超临界和亚临界分岔）。对于 2/9 < ∊ < 1/4 范围内的 ε，仅存在稳定的极限环。如果 ∊ ≽ 1/4，则无法观察到振荡解。
The simplest model of thermokinetic oscillations in a closed, chemical system requires only two first-order reaction steps (0) P → A rate = k0p, (1) A → B rate = k1(T)a. Step (0) is assumed to be thermoneutral and its rate constant to not depend on the temperature (i. e. to have zero activation energy). Step (1) is an exothermic process, and the rate constant k1 has an Arrhenius temperature dependence k1 = A1 e–E1/RT. The governing reaction rate and energy-balance equations written in dimensionless form in terms of the reduced concentration α of the intermediate A and the temperature rise θ are : dα/dז = μ e-γז ─ kαf(θ) and dθ/dז = αf(θ) ─ θ, where μ, γ and k are parameters and the function f(θ) has the form f(θ)(θ) = exp [θ/(1 + ∊θ)]. All of the dynamical behaviour of these equations can be determined qualitatively and, to leading order, quantitatively from the pool chemical approximation, γ = 0. Oscillatory solutions emerge as μ and k are varied, from points of Hopf bifurcation. These are located analytically and additional expressions are derived for calculating the growth in amplitude and period as the system moves away from these points. Both stable and unstable limit cycles (supercritical and subcritical bifurcations) can be found, provided 0 < ∊ < 2/9. For ε in the range 2/9 < ∊ < 1/4 only stable limit cycles exist. Oscillatory solutions cannot be observed if ∊ ≽ 1/4.