Diffusive boundary layer development above a sediment-water interface

A model to estimate the entry length to a fully developed diffusive boundary layer above a sediment bed, such as those found in lakes, reservoirs, rivers, and estuaries, is presented. The model is used to determine how the length of a sediment bed in mass-transfer experiments influences the measured vertical diffusive flux at the sediment-water interface. A nondimensional local mass flux is introduced in the form of a Sherwood number (Sh) and expressed as a function of both the distance from the leading edge of the sediment bed (x) and the Schmidt number (Sc). Similarly, a mean Sherwood number (Sh(ave)) for a sediment bed of length (L) is introduced. The diffusive boundary layer grows with distance, and its thickness depends on the Schmidt number (i.e., the diffusive boundary layer gets thicker and develops more quickly as the Schmidt number decreases). For Schmidt numbers greater than or equal to 100, the diffusive boundary layer begins to develop slowly but is fully developed when the nondimensional horizontal coordinate (x(+)) is approximately 1000. The Sherwood number is largest (i.e., infinity) near the leading edge of the sediment bed (i.e., at x = 0), decreases as the distance from the bed increases, and, finally, approaches a constant value for a fully developed diffusive boundary layer (Sh(infinity)). In this paper, the distance to a fully developed diffusive boundary layer (L-99) and the required length of a sediment bed are related explicitly to Sc, sheer velocity (U.), and the relative errors of local or average Sherwood numbers (Sh or Sh(ave), respectively) against the Sherwood number for the fully developed diffusive boundary layer (Sh(infinity)). The lengths L99 and L decrease as the Schmidt number increases and become independent of the Schmidt number when Sc is greater than 1000. A longer sediment bed is needed when the shear velocity or the Schmidt number is small (e.g., L-99 and L approximate to 1.0 m and 8.0 m, respectively, for Sc = 500, U-* = 0.1 cm/s, and a 3% acceptable error). Experimental studies may not be able to meet these requirements and an adjustment of measured mass-transfer rates at a sediment-water interface may be necessary. The magnitude of that adjustment is up to 50%. Its dependence on the Schmidt number, shear velocity, and bed length is given in this paper.