Reaction and diffusion in a lamellar structure: the effect of the lamellar arrangement upon yield

doi:10.1016/S0378-4371(98)00423-3

Physica A: Statistical Mechanics and its Applications

Volume 262, Issues 3–4, 15 January 1999, Pages 294-306

https://doi.org/10.1016/S0378-4371(98)00423-3 Get rights and content

Abstract

The yield of a competitive–consecutive chemical reaction in a lamellar structure is shown to be strongly influenced by the order in which the lamellae are arranged. Average measures of lamellar width, which take no account of the particular arrangement of the lamellae, perform poorly in predicting the yield. As the number of lamellae is increased, the range of possible yields grows with the number of permutations of the lamellae. The results are a warning that in order to match results from a lamellar model with experiment one needs a good deal of detailed information about the lamellae, beyond their mean width.

Introduction

Determining the yield of a chemical reaction in a fluid flow is a problem that has been of interest to many researchers [1], [2], [3], [4]. For many reactions, fluid mixing can have a significant effect on both the rate of reaction and the quality of the product [5] – the ratio between desired products and waste, formed in secondary, simultaneous (and unwanted) reactions. Clearly these issues are important for industrial chemical reaction processes, such as the Reaction Injection Moulding polymerisation process, where the reaction may fail to go to completion even if there is a stoichiometric balance of the reactants [6], [7].

In this paper we consider a model for chemical reactions between two initially segregated chemical species in a two-dimensional laminar chaotic fluid flow. This flow generates a complicated structure of thin striations of the two species [8]. The existence of these striations provides the basis for a one-dimensional ‘lamellar’ model of the fully two-dimensional problem, and so much work has been directed towards elucidating the structure of the striations [9], [10]. The interface between the chemical species remains sharp if they react with one another, and the reaction rate is taken to be infinite [11], [12]. The reaction is much harder to simulate numerically if instead the rate of reaction is finite, in which case the sharp interface between the species is replaced by a reaction zone in which the reactants are converted into products. Although there are then no sharply defined striations as such, it is useful, at least before the system becomes nearly homogeneous, to continue to use the terminology of striations and interfaces. At the early stage, it is sufficient to model the system as a series of reaction fronts at the interfaces [13], [14], [15], but as the species diffuse further across the striations, the reaction fronts meet, and interact: the full problem must then be considered. The time for this interaction to occur is governed by the rates of diffusion of the chemical species, and by the width of the striations.

Since the flow is incompressible, the striation widths are related to the length of the interface between the reactants: the thinner the striations, the longer the interface. It has been shown that the mean striation width $w ̄$ is proportional to $e^{−λt}$ at large time, $t$ , where $λ$ is the positive Lyapunov exponent of the chaotic flow [1]. The thinner the striation width, the faster the rate of reaction, and for the system we consider subsequently, the higher the yield. A detailed description of the effect of the striation width on the yield is given in [16].

Striation widths are an important ingredient in the one-dimensional lamellar model of a reaction system, in which the curved striations of the two-dimensional system are replaced by an array of exactly parallel lamellae. However, the striations produced by typical chaotic flows in both experimental and numerical studies have a wide range of widths [17], [18]. This range is a consequence of the tremendous variety of length stretches experienced at different points in the flow. For example, striation widths calculated in a recent numerical study varied over three orders of magnitude [19]. As noted in [20], the lamellar model does not provide an adequate approximation based on the mean striation width alone, and more complete information about the distribution of striation widths is needed.

Another factor that influences yield is the order in which striations of different widths are arranged [5], [20]. This aspect of the problem has received little attention hitherto. Previously, numerical simulations have assumed that the widths of adjacent striations are uncorrelated [18]. This assumption may not hold in realistic fluid flows; its influence is investigated here.

Section snippets

The reaction system

We consider the competitive–consecutive (or two-step) reaction $A+B →^{k_{1}} R, B+R →^{k_{2}} S,$ where $k_{1}$ and $k_{2}$ are the rate constants for the two reactions, and $k_{2} /k_{1} =ε⪡1 .$ As an initial state for the system we assume that the reactants $A$ and $B$ are segregated in a one-dimensional array of lamellae (see Fig. 1) and that the products $R$ (desired product) and $S$ (unwanted by-product) are initially absent.

We let $c_{A}$ , $c_{B}$ , $c_{R}$ and $c_{S}$ denote the concentrations of the corresponding chemical species in which case the system is

Numerical simulations of the one-dimensional reaction–diffusion system

In this section we describe numerical simulations of the one-dimensional reaction–diffusion system $(8)$ – $(11)$ in the interval $0⩽X⩽L,$ which initially contains lamellae of species $A$ and species $B$ of two different widths, $W>w$ . The detailed distribution of $A$ and $B$ is described for each simulation by listing the lamellae in order, with a capital letter indicating a thick lamella and a small letter indicating a thin lamella. The arrangement in Fig. 1 is thus $ABab$ . The aim is to determine the effect upon

Discussion

We have considered a competitive–consecutive chemical reaction taking place in a one-dimensional lamellar array, and in particular we have shown that when the lamellae are of different widths, the order in which they lie can significantly affect the yield of the product. As the number of lamellae increases, the range of possible yields grows with the number of permutations of the lamellae. Approximating an arrangement of lamellae by a periodic array of lamellae of mean width tends to

Acknowledgements

This work was supported by the United Kingdom Engineering and Physical Sciences Research Council (EPSRC) under its Applied Nonlinear Mathematics Programme.

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Reaction and diffusion in a lamellar structure: the effect of the lamellar arrangement upon yield

Abstract

Introduction

Section snippets

The reaction system

Numerical simulations of the one-dimensional reaction–diffusion system

Discussion

Acknowledgements

Chem. Eng. J.

Chem. Eng. Sci.

Chem. Eng. Sci.

Chem. Eng. J.

Physica A

Chem. Eng. Sci.

A fluid mechanical approach to turbulent mixing and chemical reaction. Part III: Computational and experimental results for the new micromixing model

Chem. Eng. Commun.

Modelling of rapidly mixed fast-crosslinking exothermic polymerizations

A.I.Ch.E.J.

Analysis of impingement mixing–reaction data: use of a lamellar model to generate fluid mixing information

Ind. Eng. Chem. Res.