The one-dimensional diffusion equation describes the equilibrium condition when the quantity

of sediment settling across a unit area due to the force of gravity is equal to the quantity of

sediment transported upwards resulting from the vertical component of turbulence and the

concentration gradient. The resulting equation for a given particle size is

ωc = - ε S dc / dy

(4.4)

where:

= Fall velocity of the sediment particle at a point

ω

= Concentration of particles at elevation y above the bed

c

= Exchange coefficient, also called the mass transfer coefficient, which

εS

characterizes the magnitude of the exchange of particles across any

arbitrary boundary by the turbulence

= Concentration gradient

dc/dy

= Average rate of settling of the sediment particles

ωc

= Average rate of upward sediment flow by diffusion

εS dc/dy

Integrating Equation 4.4 yields

{

}

y

c = c a exp - ω ao dy / ε s

(4.5)

where ca is the concentration of sediment with settling velocity equal to ω at a level y = a.

In order to determine the value of c at a given y, the value of ca and the variation of εs with y

ε s = βε m

(4.6)

where:

= Kinematic eddy viscosity or the momentum exchange coefficient defined by

εm

τ

εm =

(4.7)

ρ (dv / dy))

where:

= Shear stress and velocity gradient, respectively, at point y

τ and dv/dy

For two-dimensional steady uniform flow

τ = γS(y o - y) = τ o (1 - y / y o )

(4.8)

4.8

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