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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