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Heat/mass transport in shear flow over a reactive surface with inert defects
- Preyas N. Shah, Tiras Y. Lin, Eric S. G. Shaqfeh
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- Journal:
- Journal of Fluid Mechanics / Volume 811 / 25 January 2017
- Published online by Cambridge University Press:
- 13 December 2016, pp. 372-399
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We study the problem of mass transport to surfaces with heterogeneous reaction rates in the presence of shear flow over these surfaces. The reactions are first order and the heterogeneity is due to the existence of inert regions on the surfaces. Such problems are ubiquitous in the field of heterogeneous catalysis, electrochemistry and even biological mass transport. In these problems, the microscale reaction rate is characterized by a Damköhler number $\unicode[STIX]{x1D705}$, while the Péclet number $P$ is the dimensionless ratio of the bulk shear rate to the inverse diffusion time scale. The area fraction of the reactive region is denoted by $\unicode[STIX]{x1D719}$. The objective is to calculate the yield of reaction, which is directly related to the mass flux to the reactive region, denoted by the dimensionless Sherwood number $S$. Previously, we used boundary element simulations and examined the case of first-order reactive disks embedded in an inert surface (Shah & Shaqfeh J. Fluid Mech., vol. 782, 2015, pp. 260–299). Various correlations for the Sherwood number as a function of $(\unicode[STIX]{x1D705},P,\unicode[STIX]{x1D719})$ were obtained. In particular, we demonstrated that the ‘method of additive resistances’ provides a good approximation for the Sherwood number for a wide range of values of $(\unicode[STIX]{x1D705},P)$ for $0<\unicode[STIX]{x1D719}<0.78$. When $\unicode[STIX]{x1D719}\approx 0.78$, the reactive disks are in a close packed configuration where the inert regions are essentially disconnected from each other. In this work, we develop an understanding of the physics when $\unicode[STIX]{x1D719}>0.78$, by examining the inverse problem of inert disks on a reactive surface. We show that the method of resistances approach to obtain the Sherwood number fails in the limit as $\unicode[STIX]{x1D719}\rightarrow 1$, i.e. in the dilute limit of periodic inert disks, due to the existence of a surface concentration boundary layer around each disk that scales with ($1/\unicode[STIX]{x1D705}$). This boundary layer controls the screening length between inert disks and allows us to introduce a new theory, thus providing new correlations for the Sherwood number that are highly accurate in the limit of $\unicode[STIX]{x1D719}\rightarrow 1$.
Heat/mass transport in shear flow over a heterogeneous surface with first-order surface-reactive domains
- Preyas N. Shah, Eric S. G. Shaqfeh
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- Journal:
- Journal of Fluid Mechanics / Volume 782 / 10 November 2015
- Published online by Cambridge University Press:
- 08 October 2015, pp. 260-299
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Surfaces that include heterogeneous mass transfer at the microscale are ubiquitous in nature and engineering. Many such media are modelled via an effective surface reaction rate or mass transfer coefficient employing the conventional ansatz of kinetically limited transport at the microscale. However, this assumption is not always valid, particularly when there is strong flow. We are interested in modelling reactive and/or porous surfaces that occur in systems where the effective Damköhler number at the microscale can be $O(1)$ and the local Péclet number may be large. In order to expand the range of the effective mass transfer surface coefficient, we study transport from a uniform bath of species in an unbounded shear flow over a flat surface. This surface has a heterogeneous distribution of first-order surface-reactive circular patches (or pores). To understand the physics at the length scale of the patch size, we first analyse the flux to a single reactive patch. We use both analytic and boundary element simulations for this purpose. The shear flow induces a 3-D concentration wake structure downstream of the patch. When two patches are aligned in the shear direction, the wakes interact to reduce the per patch flux compared with the single-patch case. Having determined the length scale of the interaction between two patches, we study the transport to a periodic and disordered distribution of patches again using analytic and boundary integral techniques. We obtain, up to non-dilute patch area fraction, an effective boundary condition for the transport to the patches that depends on the local mass transfer coefficient (or reaction rate) and shear rate. We demonstrate that this boundary condition replaces the details of the heterogeneous surfaces at a wall-normal effective slip distance also determined for non-dilute patch area fractions. The slip distance again depends on the shear rate, and weakly on the reaction rate, and scales with the patch size. These effective boundary conditions can be used directly in large-scale physics simulations as long as the local shear rate, reaction rate and patch area fraction are known.