The Steady-State Convection-Diffusion Equation at High Péclet Numbers for a Cluster of Spheres: An Extension of Levich's Theory

Abstract

This paper describes an approximate analytical model of competitive effects between members of a dense cluster of absorbing objects, which are modeled as spheres. Neighboring absorbing spheres compete for diffusing species and thereby reduce each other's rate of absorption. Levich's well-known asymptotic (high Peźclet number) theory of convection-diffusion considers only the inner region of the concentration boundary layer; it does not describe the wake zone accurately. An extension of the Levich model is constructed for the wake zone. This is used to model intersphere competitive effects. The model demonstrates that for two neighboring spheres aligned along the flow direction, the absorption of the downstream sphere is substantially reduced vis-aź-vis the upstream sphere. The model is verified by comparison to numerical simulation studies. Both single-sphere simulations (reported in this paper) and multisphere simulations (taken from existing literature) are considered. In the single-sphere case, the discrepancy between the analytical model and the numerical results is maximally 10% at Pe = 10 and much lower at higher Peźclet numbers. An appreciable part of the error stems from the original Levich model itself, rather than from our method of extending the Levich model. In the multisphere case, the difference between the analytical model and the numerical studies is generally less than 30% . At small intersphere separations (say, center-to-center distances $<$ 5 sphere radii), the model tends to overestimate the interference effects. This is related to the fact that flow stagnancy in the space between two closely packed spheres is not taken into account in the model.