Upscaling mass adsorption and momentum transport in the crown of trees

Abstract

Airborne chemical species transport and adsorption in the crown of trees are relevant processes with many crucial environmental consequences. This multiphase and multiscale process requires incorporation of momentum transport both in the air and in the solid phases composing the biomass (leaves and branches) and coupling with species mass transport. In this work, an upscaled model for momentum transport and adsorption of chemical species in tree crowns is derived using the method of volume averaging. The model comprises four effective-medium equations, namely: the macroscopic balance equation for momentum transport in the air, which has a Darcy-like structure; the macroscopic equations for total mass and momentum transport, considering the air and deformable leaves and branches; and an unsteady upscaled equation for species mass (diffusive and convective) transport and adsorption at the surface of leaves and branches. These equations are written in terms of effective-medium coefficients that capture the essential microscale information by solving ancillary closure problems in periodic unit cells in the Laplace domain. This allowed evaluating the dynamic functionality of the unsteady adsorption and dispersion coefficients in terms of the Reynolds number, the solid velocity and the adsorption rate. The macroscopic model predictions of the average chemical species concentration were found to be in excellent agreement with direct numerical simulations (i.e., with a relative percent difference smaller than 1%), thus providing a first validation of the upscaled model. Extensions to other systems are briefly discussed.