Homogenization of a Shrinking Core Model for Gas–Solid Reactions in Granular Particles

Abstract

\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . Reactions between gases and solid particles are commonly modeled using a shrinking core framework, where a sharp interface between an inner unreacted core and an outer product shell moves inward until the reaction is complete. However, for some physical systems this sharp divide is not present, and so a better model is needed to capture a transition region for the reaction. We are interested in large particles made of many small grains where there are strong interactions between microscale granular and macroscale particulate effects, and where a shrinking core model represents behavior at the microscale level. We obtain homogenized equations for macroscale behavior by exploiting the small ratio of granular to particle lengthscales. These macroscale equations allow for a diffuse reaction front, as well as a sharp interface between reacted and unreacted solid material. We analyze the resulting model asymptotically in the limits where the reaction time is rate-limited by chemical kinetics, and separately by diffusion, determining the thickness of the reaction front. Numerical simulations support the law of additive reaction times, which states that the total reaction time is given as the sum of the conversion times under these limits. We further show how the model results can be extended to incorporate the transport of the product gas out of the solid particle, using a binary Fickian diffusion model. In metallurgical production the particle size and porosity are known to influence reaction times. Our results help quantify these effects and may be an aid in raw material selection.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . homogenization, gas-solid reactions, porous media, shrinking core model, diffusion, asymptotics \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 35B27, 35C20, 80A30, 92E20 \bfD \bfO \bfI .

Homogenised model for the electrical current distribution within a submerged arc furnace for silicon production

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Derivation of an effective thermal electrochemical model for porous electrode batteries using asymptotic homogenisation

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