Formulas for the Transportation of Bed Load

This paper introduces new formulas for the transportation of bed load that disputes Einstein’s use of his time factor, $t_E$, in calculating transport rate. His $t_E$ accounts for the falling velocity in still water of a given material, yet he applies this same $t_E$ in the context of a transport-rate problem. The authors introduce a new way of analyzing an old problem that everyone assumes needs no further development and as a result, respectfully disagree with Einstein’s use of $t_E$ within a transport-rate problem. This research presents an original formula that introduces a new time factor, $t_Z$, to be included instead. The major difference between $t_E$ and $t_Z$ is moving $v_f$ from the denominator to the numerator. The logic behind this is that the magnitude of a dense particle’s falling velocity is larger and, therefore, $t_E$ should be shorter in time. However, in the context of a transport-rate problem, it should take longer for a dense particle to be transported to its destination; thus, the introduction of $t_Z$ resulting in a longer time. With the new expression for the time factor, $t_Z$, as the settling velocity of bed-load material particles in still clear water divided by the constant of gravitational acceleration, Einstein’s bed-load transportation formula is then modified accordingly as the variables in the formula are redefined. Experimental data used by Einstein are reanalyzed, and the data clearly show that separate formulas for different specific gravity bed-load materials are needed. The modified formula for gravel grains is verified by Smart’s test data in steep channel (region of high intensity of transport) as is seen from the predicted curve running near the data points. When the bed-load transportation formulas in silty water and in differential temperature water are derived, the parameter related to the characteristics of the carrying medium is replaced accordingly. In both cases, the rate of bed-load transportation is found to be inversely proportional to the settling velocity of bed-load particles in the carrying medium. As the bed-load study involves three factors, namely, (1) characteristics of flow; (2) specific gravity of bed-load materials; and (3) characteristics of carrying medium, it can be seen that the formula derived considers these three factors in the formula modification. As Einstein is the only author in the field who brings the settling velocity of bed-load particles into the bed-load transportation study, this paper further addresses the importance of settling velocity and its role.