Abstract Derivation of macroscopic models for advection-diffusion processes in the presence of dominant heterogeneous (e.g., surface) reactions using homogenisation theory or volume averaging is often deemed unfeasible (Valdes-Parada et al., 2011; Battiato and Tartakovsky, 2011a) due to the strong coupling between scales that characterise such systems. In this work, we show how the upscaling can be carried out by applying and extending the methods presented in Allaire and Raphael (2007), Mauri (1991). The approach relies on the decomposition of the microscale concentration into a reactive component, given by the eigenfunction of the advection-diffusion operator, the associated eigenvalue which represents the macroscopic effective reaction rate, and a non-reactive component. The latter can be then upscaled with a two-scale asymptotic expansion and the final macroscopic equation is obtained for the leading order. The same method can also be used to overcome another classical assumption, namely of non solenoidal velocity fields, such as the case of deposition of charged colloidal particles driven by electrostatic potential forces. The whole upscaling procedure, which consists in solving three cell problems, is implemented for arbitrarily complex two- and three-dimensional periodic structures using the open-source finite volume library OpenFOAM®. We provide details on the implementation and test the methodology for two-dimensional periodic arrays of spheres, and we compare the results against fully resolved numerical simulations, demonstrating the accuracy and generality of the upscaling approach. The effective velocity, dispersion and reaction coefficients are obtained for a wide range of Peclet and surface Damkohler numbers, and for Coulomb-like forces to the grains. Noticeably, all the effective transport parameters are significantly different from the non-reactive (conserved scalar) case, as the heterogeneity introduced by the reaction strongly affects the micro-scale profiles.

中文翻译：

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多孔介质中过滤和非均相反应的宏观模型

摘要 使用均质化理论或体积平均法在存在显性异质（例如表面）反应的情况下推导对流扩散过程的宏观模型通常被认为是不可行的（Valdes-Parada 等，2011；Battiato 和 Tartakovsky，2011a），因为表征此类系统的尺度之间的强耦合。在这项工作中，我们展示了如何通过应用和扩展 Allaire 和 Raphael（2007）、Mauri（1991）中提出的方法来进行升级。该方法依赖于将微观浓度分解为反应成分，由对流扩散算子的特征函数、表示宏观有效反应速率的相关特征值和非反应成分给出。然后可以通过两尺度渐近展开对后者进行放大，并获得领先阶次的最终宏观方程。同样的方法也可以用来克服另一个经典假设，即非螺线管速度场，例如由静电势力驱动的带电胶体粒子沉积的情况。整个放大过程包括解决三个单元问题，使用开源有限体积库 OpenFOAM® 为任意复杂的二维和三维周期结构实现。我们提供了关于二维周期性球体阵列的实现和测试方法的详细信息，并将结果与​​完全解析的数值模拟进行了比较，证明了放大方法的准确性和通用性。有效速度，色散和反应系数是针对大范围的 Peclet 和表面 Damkohler 数以及对晶粒的类库仑力获得的。值得注意的是，所有有效传输参数都与非反应性（守恒标量）情况显着不同，因为反应引入的异质性强烈影响微尺度剖面。