A bed-load formula based on kinetic theory

Bed-load plays an important role in sediment transport in fluvial rivers. This paper presents a bed-load formula on the basis of kinetic theory. The pick-up rate of bed-load sediment in the formula is obtained by integrating the velocity distribution function derived by solving the Boltzmann equation of kinetic theory. During the integration, the effect of incipient probability is considered. The bed-load formula is derived by substituting the expression of the pick-up rate into the basic equation of bed-load transport developed by Einstein. Comparisons of the simulations with available experimental data for both weak and intensive sediment transport are presented. The relative errors range from 10% to 50%, which illustrates that the new formula can reasonably represent the statistical characteristics of the bed-load particles moving on the bed surface. Furthermore, the new formula has remarkably improved the result of Einstein equation for intensive sediment transport, and is very similar to the classical bed-load formulas, which proves the validity of the theoretical derivation of the formula.