Dimensional Analysis of Reaeration Rate in Streams

Atmospheric reaeration at the free surface of lakes and streams is a relevant process for water quality, thus the amount of oxygen transferred to the water body should be carefully estimated. Recent studies have demonstrated that available equations for estimation of the reaeration rate offer a poor fit with field data different from those for which each equation was originally developed. Thus, none of the available equations is applicable to all stream hydrodynamic conditions; on the contrary, they remain stream-specific, probably since some parameters involved in the process have been neglected in their formulation and their expressions are too simplistic. This paper proposes a comprehensive approach to the mass-transfer process at the air-water interface that is based on dimensional analysis. Careful inspection of equations in the literature shows that the mass-transfer process at the air-water interface has been affected by 14 different parameters. The application of dimensional analysis produces, for a wide rectangular section if wind speed is negligible, a dimensionless equation for the mass-transfer rate, where this rate is a function of the Froude number, channel slope, Reynolds number, Sherwood number, Weber number, and relative roughness. This expression is further developed to address the reaeration process in streams and rivers. As a result, at a fixed temperature, the dimensionless reaeration rate $K_{a\mathrm{ND}}$ (where ND denotes nondimensional) is finally a function of only the Froude number, channel slope, Reynolds number, and relative roughness. Moreover, the application of the Darcy-Wiesbach equation allows this dimensionless rate $K_{a\mathrm{ND}}$ to be considered as a function of only three of the aforementioned parameters. This result provides a comprehensive approach to the reaeration process that can also explain the unreliability of the literature equations available up to now.