The number N^ of particles of size D per unit area of the bed surface 

 that at any instance become free to move and indeed do move is propor- 

 tional to this probability as well as to the total population of similar 

 particles per unit bed surface. It is evident that 



A X D 2 



(4-14) 



where Ai is a constant. 



The rate of transportation q B which is defined as the rate in dry 

 weight at which solid particles move across a section of unit width 

 oriented perpendicular to the flow can be expressed as 



% = N 1 W 1 V 1 = -S W 1 V 1 (4 - 15 > 



V 



where W^ = A 2 D Y s , Ys being the dry unit weight of the particles and 

 Vi the speed of the particle propagation. It should be remembered that 

 the motion of the particles is not continuous, but that it consists of a 

 sequence of discrete steps. It will be reasonable, therefore, to express 

 this speed of propagation as the ratio of the average distance covered by 

 a particle in a step and of the total time required for the completion of 

 a full cycle of motion. If we denoted the former by I and the latter by 

 t equation (4-15) after the substitution for Wl becomes 



p 3 -0 A I 



% = 2~ A 2 D y s t = Jl pDYs t (4-16) 



AjD Ai 



The distance & as in the case of a steady mean flow may be considered as 

 a charactsristic length proportional to the grain diameter, so that X= AD. 

 Moreover by definition t can be written as 



t = t ± + t 2 (4-17) 



where t\ is the part of the cycle during which the particle is at rest 

 (between two consecutive excursions) and t2 is the part during which the 

 particle is in motion. Experiments with light-weight coarse material and 

 steady mean flows in flumes have demonstrated that in general t 2 is much 

 smaller than t\ which led to the conclusion that for all practical pur- 

 poses we may assume t = t\. Einstein (1950) has defined t\, which he 

 called "the exchange time" as a measure of the time required for the 

 replacement of a particle that is just being picked up by the flow at a 

 certain spot of the bed by a similar particle that is being brought to 

 rest at the same spot. Therefore, one may think of ti as a parameter 

 which depends on the properties of the particle and the surrounding fluid 

 only and which, consequently, is independent of the local flow conditions. 



16 



