Diffusion-controlled first-order surface reaction in turbulent flow
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L, NONLINEAR, AND SOFT MATTER PHYSICS
Diffusion-Controlled First-Order Surface Reaction in Turbulent Flow E. G. Obrazovskiœ OOO Khimpolitekh, Novosibirsk, 630060 Russia Novosibirsk State University, Novosibirsk, 630090 Russia e-mail: [email protected] Received January 24, 2006
Abstract—The effect of advective diffusion on the rate of reactant consumption by a first-order surface reaction is analyzed in the fast-reaction limit. The decay of reactant concentration is described by the function n(t) ~ exp(–λt). In the limit of well-developed turbulence, the scaling estimates λ ~ L–1κ3/4µ1/4 and λ ~ fκ3/4µ1/4 are obtained, respectively, for a confined flow with characteristic length scale L and in the case when the reactants are contained near the surface by an external field with potential U/T = fx, where κ is molecular diffusivity and µ is the constant parameter in the eddy diffusivity Dadv = µx4 (x is distance to the wall). The coefficients in the scaling laws are evaluated by a variational method and by numerical solution of the governing equations. PACS numbers: 47.70.Fw, 47.27.Qb DOI: 10.1134/S1063776106070132
1. INTRODUCTION Bulk reaction kinetics are controlled by diffusion if the rate of chemical transformation of the reactants is sufficiently high. Advective diffusion in turbulent flow enhances the overall reaction rate through faster mixing of reactants. This process is generally modeled by passive scalar advection [1, 2]; i.e., the effect of the scalar field on the flow is neglected. One typical example is well-developed turbulence [3]. Another important example is the elastic turbulence developing as a result of flow instability in dilute polymer solutions [4]. Considerable progress in analytical description of turbulent flows has been achieved by using the fact that these flows are locally laminar [5–7]. At the final stage of the mixing process, the inhomogeneities of the scalar concentration are mainly localized in the near-wall boundary layer, where mixing is less efficient [8, 9]. The passive-scalar statistics and time evolution controlled by the universal structure of the fluctuating boundary-layer velocity field in welldeveloped or elastic turbulence were also analyzed in those studies. In [10], it was shown that the results obtained for passive scalar evolution in turbulence can be generalized to describe a fast (bulk) chemical reaction. Here, the case of surface reaction is considered. Numerous applications of heterogeneous surface reactions include electrode reactions, electroplating, dissolution processes [11], and membrane separation [12]. The overall rate of reactant consumption by a surface reaction is controlled by diffusion of reactants to the surface. Heterogeneous surface reactions have been
well studied only for regimes controlled by molecular diffusion and for laminar flow over the surface [11]. This study is focused on surface reaction in turbulent flow, where reactants are brought to the surface by molecular and turbulent diffusion. Therefore, the results obtained in [8, 9] can be
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